Finite Gorenstein Injective Dimension Research Articles

Gorenstein cohomology of N-complexes

Let [Formula: see text] and [Formula: see text] be [Formula: see text]-complexes with [Formula: see text] an integer such that [Formula: see text] has finite Gorenstein projective dimension and [Formula: see text] has finite Gorenstein injective dimension. We define the [Formula: see text]th Gorenstein cohomology groups [Formula: see text] [Formula: see text] via a strict Gorenstein precover [Formula: see text] of [Formula: see text] and a strict Gorenstein preenvelope [Formula: see text] of [Formula: see text]. Using Gaussian binomial coefficients we show that there exists an isomorphism [Formula: see text] which extends the balance result of Liu [Relative cohomology of complexes. J. Algebra 502 (2018) 79–97] to the [Formula: see text]-complex case.

Linkage of modules with respect to a semidualizing module

The notion of linkage with respect to a semidualizing module is introduced. It is shown that over a Cohen-Macaulay local ring with canonical module, every Cohen-Macaulay module of finite Gorenstein injective dimension is linked with respect to the canonical module. For a linked module $M$ with respect to a semidualizing module, the connection between the Serre condition $(S_n)$ on $M$ with the vanishing of certain local cohomology modules of its linked module is discussed.

Relative cohomology of complexes

Let R be an arbitrary ring and C a complex with finite Gorenstein projective dimension (that is, the supremum of Gorenstein projective dimension of all R-modules in C is finite). For each complex D, we define the nth relative cohomology group ExtGPn(C,D) by the equality ExtGPn(C,D)=HnHom(G,D), where G⟶C is a strict Gorenstein projective precover of C. If D is a complex with finite Gorenstein injective dimension (that is, the supremum of Gorenstein injective dimension of all R-modules in D is finite), then one can use a dual argument to define a notion of relative cohomology group ExtGIn(C,D). We show that if C is a complex with finite Gorenstein projective dimension and D a complex with finite Gorenstein injective dimension, then for each n∈Z there exists an isomorphism ExtGPn(C,D)≅ExtGIn(C,D). This shows that the relative cohomology functor of complexes is balanced. Some induced exact sequences concerning relative cohomology groups are considered.

When do Foxby classes coincide with the classes of modules of finite Gorenstein dimensions?

The relation between the Auslander (resp., Bass) class and the class of modules with finite Gorenstein projective (resp., injective) dimension is well known when these mentioned classes are built with a dualizing module over Noetherian n-perfect rings. Basically, the results are necessary conditions to ensure that both classes coincide. In this article we try to extend and sometimes improve some of these results by weakening the condition of being dualizing. Among other results, we prove that a Wakamatsu tilting module with some extra conditions is precisely a module RC such that the Bass class BC(R) coincides with the class of modules of finite Gorenstein injective dimension.

Gorenstein Injective Envelopes of Artinian Modules

Let R be a commutative Noetherian ring and A an Artinian R-module. We prove that if A has finite Gorenstein injective dimension, then A possesses a Gorenstein injective envelope which is special and Artinian. This, in particular, yields that over a Gorenstein ring any Artinian module possesses a Gorenstein injective envelope which is special and Artinian.

GORENSTEIN INJECTIVE MODULES AND A GENERALIZATION OF ISCHEBECK FORMULA

Let (R,[Formula: see text]) be a commutative Noetherian local ring and let M and N be nonzero finitely generated R-modules of finite injective dimension and finite Gorenstein injective dimension, respectively. In this paper, we prove a generalization of Ischebeck formula, that is [Formula: see text].

Dual of the Auslander-Bridger formula and GF-perfectness

Ext-finite modules were introduced and studied by Enochs and Jenda. We prove under some conditions that the depth of a local ring is equal to the sum of the Gorenstein injective dimension and Tor-depth of an Ext-finite module of finite Gorenstein injective dimension. Let $(R,\mathfrak m)$ be a local ring. We say that an $R$-module $M$ with $\dim_R M=n$ is a Grothendieck module if the $n$-th local cohomology module of $M$ with respect to $\mathfrak m$, $\mathrm{H}_{\mathfrak m}^n (M)$, is non-zero. We prove the Bass formula for this kind of modules of finite Gorenstein injective dimension and of maximal Krull dimension. These results are dual versions of the Auslander-Bridger formula for the Gorenstein dimension. We also introduce GF-perfect modules as an extension of quasi-perfect modules introduced by Foxby.

A Bass Formula for Gorenstein Injective Dimension

In this article a generalized version of the Bass formula is proved for finitely generated modules of finite Gorenstein injective dimension over a commutative Noetherian ring.

A Generalization of a Theorem of Bass

Let (R, 𝔪) be a commutative Noetherian local ring. In this paper we prove that if R admits a finite module of finite Gorenstein injective dimension and maximal Krull dimension then R is Cohen–Macaulay. This generalizes a theorem of Bass and improves a recent result of Takahashi where the ring is assumed to have a dualizing complex.

The existence of finitely generated modules of finite Gorenstein injective dimension

In this note, we study commutative Noetherian local rings having finitely generated modules of finite Gorenstein injective dimension. In particular, we consider whether such rings are Cohen-Macaulay.

Cohomology theories based on Gorenstein injective modules

In this paper we study relative and Tate cohomology of modules of finite Gorenstein injective dimension. Using these cohomology theories, we present variations of Grothendieck local cohomology modules, namely Gorenstein and Tate local cohomology modules. By applying a sort of Avramov-Martsinkovsky exact sequence, we show that these two variations of local cohomology are tightly connected to the generalized local cohomology modules introduced by J. Herzog. We discuss some properties of these modules and give some results concerning their vanishing and non-vanishing.

Gorenstein Injective Dimension and Dual Auslander–Bridger Formula#

An R-module M satisfies the dual Auslander–Bridger formula if Gid R M + Tor − depth M = depth R. Let R be a complete Cohen–Macaulay local ring with residue field k and M be a non-injective Ext-finite R-module such that for all i ≥ 1 and all indecomposable injective R-modules E ≠ E(k). If M has finite Gorenstein injective dimension, then we will prove that M satisfies the dual Auslander–Bridger formula if either Ext-depth M > 0 or Ext-depth M = 0 and Gid R M ≠ 1. We denote Gorenstein injective envelope of M by G(M). If R is a Gorenstein local ring and M is a non-Gorenstein injective finitely generated R-module and G(M) is reduced, then this formula holds for and its cosyzygy. As the last result, if R is a regular local ring, then dual Auslander–Bridger holds for any non-injective Ext-finite R − module.

Gorenstein injective dimension and Tor-depth of modules

The main aim of this paper is to obtain a dual result to the now well known Auslander-Bridger formula for G-dimension. We will show that if R is a complete Cohen-Macaulay ring with residue field k, and M is a non-injective h-divisible Ext-finite R-module of finite Gorenstein injective dimension such that for each $i \geq 1$ Ext i (E,M) = 0 for all indecomposable injective R-modules $E \neq E(k)$ , then the depth of the ring is equal to the sum of the Gorenstein injective dimension and Tor-depth of M. As a consequence, we get that this formula holds over a d-dimensional Gorenstein local ring for every nonzero cosyzygy of a finitely generated R-module and thus in particular each such n th cosyzygy has its Tor-depth equal to the depth of the ring whenever $n \geq d$ .