Totally acyclic complexes over noetherian schemes

We define a notion of total acyclicity for complexes of flat quasi-coherent sheaves on a semi-separated noetherian scheme, and study these complexes using the pure derived category of flat quasi-coherent sheaves. We prove that a scheme is Gorenstein if and only if every acyclic complex of flat quasi-coherent sheaves is totally acyclic. Our formalism also removes the need for a dualising complex in several known results for rings, including Jørgensen's proof of the existence of Gorenstein projective precovers.     

Gorenstein injective envelopes and covers over two sided noetherian rings

We prove that the class of Gorenstein injective modules is both enveloping and covering over a two sided noetherian ring such that the character modules of Gorenstein injective modules are Gorenstein flat. In the second part of the paper we consider the connection between the Gorenstein injective modules and the strongly cotorsion modules. We prove that when the ring R is commutative noetherian of finite Krull dimension, the class of Gorenstein injective modules coincides with that of strongly cotorsion modules if and only if the ring R is in fact Gorenstein.     

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.     

关于Gorensteincotorsion模

讨论了Gorensteincotorsion模与内射模之间的关系,证明了R是GorensteinvonNeumann正则环当且仅当任意R模M的Oorensteincotorsion包络与内射包络是同构的,当且仅当E（M）／M是Gorenstein平坦模,同时,也讨论了Gorensteincotorsion模与cotorsion模之间的联系。     

Injective Envelopes and (Gorenstein) Flat Covers

We characterize left Noetherian rings in terms of the duality property of injective preenvelopes and flat precovers. For a left and right Noetherian ring R, we prove that the flat dimension of the injective envelope of any (Gorenstein) flat left R-module is at most the flat dimension of the injective envelope of _R R. Then we get that the injective envelope of _R R is (Gorenstein) flat if and only if the injective envelope of every Gorenstein flat left R-module is (Gorenstein) flat, if and only if the injective envelope of every flat left R-module is (Gorenstein) flat, if and only if the (Gorenstein) flat cover of every injective left R-module is injective, and if and only if the opposite version of one of these conditions is satisfied.     

On Gorenstein injective discrete modules over profinite groups

Gorenstein homological algebra was introduced in categories of modules. But it has proved to be a fruitful way to study various other categories such as categories of complexes and of sheaves. In this paper, the research of relative homological algebra in categories of discrete modules over profinite groups is initiated. This seems appropriate since (in some sense) the subject of Gorenstein homological algebra had its beginning with Tate homology and cohomology over finite groups. We prove that if the profinite group has virtually finite cohomological dimension then every discrete module has a Gorenstein injective envelope, a Gorenstein injective cover and we study various cohomological dimensions relative to Gorenstein injective discrete modules. We also study the connection between relative and Tate cohomology theories.     

Gorenstein Injective and Gorenstein Flat Resolution of Modules Over Gorenstein Rings

Abstract

Let R be a commutative Noetherian ring. There are several characterizations of Gorenstein rings in terms of classical homological dimensions of their modules. In this paper, we use Gorenstein dimensions (Gorenstein injective and Gorenstein flat dimension) to describe Gorenstein rings. Moreover a characterization of Gorenstein injective (resp. Gorenstein flat) modules over Gorenstein rings is given in terms of their Gorenstein flat (resp. Gorenstein injective) resolutions.     

Gorenstein dimensions of unbounded complexes and change of base (with an appendix by Driss Bennis)

For a commutative ring R and a faithfully flat R-algebra S we prove,under mild extra assumptions,that an R-module M is Gorenstein flat if and only if the left S-module S?_R M is Gorenstein flat,and that an R-module N is Gorenstein injective if and only if it is cotorsion and the left S-module Hom R(S,N)is Gorenstein injective.We apply these results to the study of Gorenstein homological dimensions of unbounded complexes.In particular,we prove two theorems on stability of these dimensions under faithfully flat (co-)base change.     

Gorenstein categories and Tate cohomology on projective schemes

We study Gorenstein categories. We show that such a category has Tate cohomological functors and Avramov–Martsinkovsky exact sequences connecting the Gorenstein relative, the absolute and the Tate cohomological functors. We show that such a category has what Hovey calls an injective model structure and also a projective model structure in case the category has enough projectives. As examples we show that if X is a locally Gorenstein projective scheme then the category ??????(X) of quasi‐coherent sheaves on X is such a category and so has these features. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stable local cohomology

For a module having a complete injective resolution, we define a stable version of local cohomology. This gives a functor to the stable category of Gorenstein injective modules. We show that in many ways this functor behaves like the usual local cohomology functor. Our main result is that when there is only one nonzero local cohomology module, there is a strong connection between that module and the stable local cohomology module; in fact, the latter gives a Gorenstein injective approximation of the former.     

n-Strongly Gorenstein Projective,Injective and Flat Modules

In this article, we study the relation between m-strongly Gorenstein projective (resp., injective) modules and n-strongly Gorenstein projective (resp., injective) modules whenever m ≠ n, and the homological behavior of n-strongly Gorenstein projective (resp., injective) modules. We introduce the notion of n-strongly Gorenstein flat modules. Then we study the homological behavior of n-strongly Gorenstein flat modules, and the relation between these modules and n-strongly Gorenstein projective (resp., injective) modules.     

Transfer of Gorenstein dimensions along ring homomorphisms

A central problem in the theory of Gorenstein dimensions over commutative noetherian rings is to find resolution-free characterizations of the modules for which these invariants are finite. Over local rings, this problem was recently solved for the Gorenstein flat and the Gorenstein projective dimensions; here we give a solution for the Gorenstein injective dimension. Moreover, we establish two formulas for the Gorenstein injective dimension of modules in terms of the depth invariant; they extend formulas for the injective dimension due to Bass and Chouinard.     

Gorenstein Dimensions over Ring Homomorphisms

In this paper, we study injective dimension and Gorenstein injective dimension over local ring homomorphisms. Some well-known results are generalized. For example, the Bass formula for Gorenstein injective dimension of complexes is extended. As applications, some characterizations of Gorenstein rings are obtained.     

On gorenstein injective and projective comodules

Gorenstein injective, projective and coflat comodules were introduced and studied by Asensio and Enochs et al. We further investigate these comodules and introduce n-Gorenstein injective, projective and coflat comodules, which give a new characterization of Gorenstein comodules.     

Strict Mittag–Leffler Conditions on Gorenstein Modules

In this article, we investigate the relations between Gorenstein projective modules and Gorenstein flat modules in terms of strict Mittag–Leffler condition. We give some conditions under which Gorenstein projectives are Gorenstein flat, and discuss when the direct limits of Gorenstein projective modules are Gorenstein flat. Moreover, we study the dual of Gorenstein injective modules with strict Mittag–Leffler condition.     

Strongly Gorenstein projective, injective, and flat modules

In this paper, we study a particular case of Gorenstein projective, injective, and flat modules, which we call, respectively, strongly Gorenstein projective, injective, and flat modules. These last three classes of modules give us a new characterization of the first modules, and confirm that there is an analogy between the notion of “Gorenstein projective, injective, and flat modules” and the notion of the usual “projective, injective, and flat modules”.     

ω-Gorenstein injective modules and dimensions

Gorenstein injective modules and dimensions have been studied extensively by many authors. In this paper, we investigate Gorenstein injective modules and dimensions relative to a Wakamatsu tilting module.     

ω-Gorenstein Modules

We introduce the notion of ω-Gorenstein modules, where ω is a faithfully balanced self-orthogonal module. This gives a common generalization of both Gorenstein projective modules and Gorenstein injective modules. We consider such modules in the tilting theory. Consequently, some results due to Auslander and colleagues and Enochs and colleagues are generalized.     

A Generalization of Gorenstein Injective and Flat Mo dules

In this article, we introduce and study the concept of n-Gorenstein injective(resp., n-Gorenstein flat) modules as a nontrivial generalization of Gorenstein injective(resp., Gorenstein flat) modules. We investigate the properties of these modules in various ways. For example, we show that the class of n-Gorenstein injective(resp., n-Gorenstein flat) modules is closed under direct sums and direct products for n ≥ 2. To this end, we first introduce and study the notions of n-injective modules and n-flat modules.