Published October 27, 2026
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Obstruction modules of non-generic Eagon–Northcott complexes: explicit homology decompositions and spectral-sequence degeneration
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Description
For the generic p×q matrix the Eagon–Northcott complex is a minimal free resolution of the
maximal-minors ideal, and its Betti table is classical. When the generic matrix is specialized to
a structured pencil, the specialized complex may fail to be acyclic. Its homology modules Di =
TorS i (R, M) are the obstruction modules of the title: they measure, as graded modules supported
on coordinate subspaces, the failure of acyclicity of the specialized Eagon–Northcott complex.
We compute Di completely for the rank-one pencil A = x1B and for the decomposable diagonal
pencil A = diag(x1, . . . , xp, 0, . . . , 0); for the latter we give a complete decomposition of every Di
into submodules supported on the distinct coordinate subspaces V (xt : t ∈ T). For the weighted
diagonal variant with an arbitrary multiplicity map v : [p] → [s], the first obstruction module is
L ∅=S⊆[p] (R/(xv(r) : r ∈ S))( q | − S| p ) (−p), which interpolates between the diagonal and rank-one
families. We also treat the full-column augmentation A = diag(x1, . . . , xp, 0, 0) + xp+11e T p+1
of the diagonal pencil: here Ip(A) is the squarefree monomial ideal generated by all degree
p products of the p + 1 variables, H1 is the direct sum of the R/(xi , xj )(−p), and all higher
obstruction modules vanish. For the full-column pencil with several appended zero columns
we reduce to the truncated complexes C [t] (1 ≤ t ≤ p + 1) and compute each Hn(C [t] ) as the
middle term of a short exact sequence whose outer terms are direct sums of the cyclic modules
R/(xs : s ∈ S); for r ≤ p + 1 this yields the corresponding decomposition of Hi(F(A)). The
change-of-rings spectral sequence Epq 2 = TorR p (k, Dq) ⇒ TorS p+q (k, M) expresses the generic
Betti table as a filtration whose graded pieces are subquotients of TorR • (k, Dq), and yields
an upper bound on the regularity of R/Ip(A); this bound is sharp on the principal stratum.
The q = 0 row records the exact Betti table of the specialized quotient; when the spectral
sequence degenerates at E2 , as it does for the computed families, this row also identifies the
corresponding associated-graded pieces of the generic Betti table. We compare these bounds
with the generic linearization predictions of Cabarcas–Gaggero–Gorla [7] and with the sparse
determinantal resolutions of Boocher [9]. The main results are the explicit computation of the
obstruction modules Di and of the E2 -degeneration for structured non-generic pencils, together
with the quantitative collapse of the free resolution in explicit parameter ranges.