Notes on endomorphisms, local cohomology and completion

Let M M denote a finitely generated module over a Noetherian ring R R . For an ideal I ⊂ R I \subset R there is a study of the endomorphisms of the local cohomology module H I g ( M ) , g = g r a d e ( I , M ) , H^g_I(M), g = grade(I,M), and related results. Another subject is the study of left derived functors of the I I -adic completion Λ i I ( H I g ( M ) ) \Lambda ^I_i(H^g_I(M)) , motivated by a characterization of Gorenstein rings given in [25]. This provides another Cohen-Macaulay criterion. The results are illustrated by several examples. There is also an extension to the case of homomorphisms of two different local cohomology modules.

Injective stabilization of additive functors, III. Asymptotic stabilization of the tensor product

The injective stabilization of the tensor product is subjected to an iterative procedure that utilizes its bifunctor property. The limit of this procedure, called the asymptotic stabilization of the tensor product, provides a homological counterpart of Buchweitz's asymptotic construction of stable cohomology. The resulting connected sequence of functors is isomorphic to Triulzi's J-completion of the Tor functor. A comparison map from Vogel homology to the asymptotic stabilization of the tensor product is constructed and shown to be always epic. The category of finitely presented functors is shown to be complete and cocomplete. As a consequence, the inert injective stabilization of the tensor product with fixed variable a finitely generated module over an artin algebra is shown to be finitely presented. Its defect and consequently all right-derived functors are determined. New notions of asymptotic torsion and cotorsion are introduced and are related to each other.

On Proper and Exact Relative Homological Dimensions

In Enochs’ relative homological dimension theory occur the (co)resolvent and (co)proper dimensions, which are defined by proper and coproper resolutions constructed by precovers and preenvelopes, respectively. Recently, some authors have been interested in relative homological dimensions defined by just exact sequences. In this paper, we contribute to the investigation of these relative homological dimensions. First we study the relation between these two kinds of relative homological dimensions and establish some transfer results under adjoint pairs. Then relative global dimensions are studied, which lead to nice characterizations of some properties of particular cases of self-orthogonal subcategories. At the end of this paper, relative derived functors are studied and generalizations of some known results of balance for relative homology are established.

Localization, Isomorphisms and Adjoint Isomorphism in the Category Comp(A - Mod)

A and B are considered to be non necessarily commutative rings and X a complex of (A - B) bimodules. The aim of this paper is to show that: The functors \overline{EXT}^n_{Comp(A-Mod)}(X,-): Comp(A-Mod) \longrightarrow Comp(B-Mod) and Tor_n^{Comp(B-Mod)}(X,-): Comp(B-Mod) \longrightarrow Comp(A-Mod) are adjoint functors. The  functor S_C^{-1}() commute with  the functors X\bigotimes - , Hom^{\bullet}(X,-) and their corresponding derived functors  \overline{EXT}^n_{Comp(A-Mod)}(X,-) and  Tor_n^{Comp(B-Mod)}(X,-).