Completed Functor 𝑆 −1 ̂() of the Localization Functor 𝑆 −1 (), Isomorphism and Adjunction

Abstract: This article serves as a continuation of our previous work 1, which remains our primary reference for investigating specific homological properties with completion. Let the rings not be necessarily commutative and the modules be the unitary left (resp. right) modules. Let (𝑮, (𝑮𝒏 )𝒏∈ℕ) be a filtered normal group equipped with the group topology associated with the filtration (𝑮𝒏 )𝒏∈ℕ formed of normal subgroups and 𝓒(𝑮) the set of Cauchy sequences with values in 𝑮. We define an equivalence relation 𝓡 on 𝓒(𝑮) by: (𝒙𝒏 )𝓡(𝒚𝒏 ) ⇔ (𝒙𝒏 ) − (𝒚𝒏 ) = (𝒙𝒏 − 𝒚𝒏 ) converges to 0, noted by (𝒙𝒏 − 𝒚𝒏 ) → 𝟎. The quotient set 𝓒(𝑮)/𝓡: = {(𝒙𝒏 ̂) ∣ (𝒙𝒏 ) ∈ 𝓒(𝑮)} denoted 𝑮̂ is equipped with a group structure and is called the completed groupe of 𝑮. For any filtered ring (resp. left 𝑨-module) (𝑨, (𝑰𝒏 )𝒏∈ℕ) (resp. (𝑴, (𝑴𝒏 )𝒏∈ℕ) ), the completed group 𝑨̂ (resp. 𝑴̂ ) is equipped with a ring structure (resp. 𝑨̂-module) by (𝒂𝒏 ̂) ×̂ (𝒃𝒏 ̂) = (𝒂𝒏𝒃𝒏 ̂) (𝒓𝒆𝒔𝒑. (𝒂𝒏̂) ⋅ (𝒎𝒏 ̂) = (𝒂𝒏 ⋅ 𝒎𝒏 ̂ )) where (𝒂𝒏 ̂), (𝒃𝒏 ̂) ∈ 𝑨̂ (resp. (𝒎𝒏 ̂) ∈ 𝑴̂ ) called completed ring (resp. module) of 𝑨 (resp. 𝑴 ). And for all saturated multiplicative subset 𝑺 of 𝑨 that satisfies the left Ore conditions, 𝑺̂ = {(𝒙𝒏 ̂) ∈ 𝑨̂ ∣ (𝒙𝒏 ̂) ≠ 𝟎̂ and ∃𝒏𝟎 ∈ ℕ, 𝒏 ≥ 𝒏𝟎, 𝒙𝒏 ∈ 𝑺} is a saturated multiplicative subset of 𝑨̂ that satisfies the left Ore conditions 1. Among the main results of this article, we have : - the functors 𝑺 −𝟏 ̂() is isomorphic to 𝑺̂−𝟏 (𝑨̂) ⊗𝑨̂−. and 𝑺̂−𝟏 () is isomorphic to 𝑺 −𝟏̂(𝑨) ⊗𝑨̂−. - the functors 𝑯𝒐𝒎𝑨̂(𝑺̂−𝟏𝑨 ⊗𝑨̂ 𝑴̂ , −) and 𝑯𝒐𝒎𝑨̂(𝑺 −𝟏𝑨̂⊗𝑨 𝑴, −) are isomorphic. - the functors 𝑺 −̂𝟏𝑨 ⊗𝑨 - and 𝑯𝒐𝒎𝑨̂(𝑺̂−𝟏𝑨̂, −) are adjoints.