Cohen–Macaulay Modules and Holonomic Modules Over Filtered Rings

We study Gorenstein dimension and grade of a module M over a filtered ring whose associated graded ring is a commutative Noetherian ring. An equality or an inequality between these invariants of a filtered module and its associated graded module is the most valuable property for an investigation of filtered rings. We prove an inequality G?dim M ≤ G?dim gr M and an equality grade M = grade gr M, whenever Gorenstein dimension of gr M is finite (Theorems 2.3 and 2.8). We would say that the use of G-dimension adds a new viewpoint for studying filtered rings and modules. We apply these results to a filtered ring with a Cohen–Macaulay or Gorenstein associated graded ring and study a Cohen–Macaulay, perfect, or holonomic module.     

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 tilting pairs

Abstract

Let A be an n-Gorenstein ring. Employing the theory developed by Enochs on the existence of Gorenstein preenvelopes and precovers, we introduce the concept of Gorenstein tilting pair. Moreover, we give a simple characterization on Gorenstein tilting pair, which shows that Gorenstein cotilting and tilting modules are special examples of Gorenstein tilting pair.     

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.     

关于Gorenstein投射维数的一个注记

本文研究了Gorenstein投射维数的相关问题.利用经典同调维数的研究方法,给出了Gorenstein投射维数有限模的Gorenstein投射维数的一个刻画,并利用这一结果证明了Gorenstein完全环和Artin环的Gorenstein整体维数分别由各自的循环模和单模的Gorenstein投射维数来确定.这些结论丰富了Gorenstein同调代数理论.     

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.     

Gorenstein Flat Covers and Gorenstein Cotorsion Modules Over Integral Group Rings

Every module over an Iwanaga–Gorenstein ring has a Gorenstein flat cover [13] (however, only a few nontrivial examples are known). Integral group rings over polycyclic-by-finite groups are Iwanaga–Gorenstein [10] and so their modules have such covers. In particular, modules over integral group rings of finite groups have these covers. In this article we initiate a study of these covers over these group rings. To do so we study the so-called Gorenstein cotorsion modules, i.e. the modules that split under Gorenstein flat modules. When the ring is ℤ, these are just the usual cotorsion modules. Harrison [16] gave a complete characterization of torsion free cotorsion ℤ-modules. We show that with appropriate modifications Harrison's results carry over to integral group rings ℤG when G is finite. So we classify the Gorenstein cotorsion modules which are also Gorenstein flat over these ℤG. Using these results we classify modules that can be the kernels of Gorenstein flat covers of integral group rings of finite groups. In so doing we necessarily give examples of such covers. We use the tools we develop to associate an integer invariant n with every finite group G and prime p. We show 1≤n≤|G : P| where P is a Sylow p-subgroup of G and gives some indication of the significance of this invariant. We also use the results of the paper to describe the co-Galois groups associated to the Gorenstein flat cover of a ℤG-module. Presented by A. Verschoren Mathematics Subject Classifications (2000) 20C05, 16E65.     

Finiteness conditions in the stable module category

We study groups whose cohomology functors commute with filtered colimits in high dimensions. We relate this condition to the existence of projective resolutions which exhibit some finiteness properties in high dimensions, and to the existence of Eilenberg–Mac Lane spaces with finitely many n-cells for all sufficiently large n. To that end, we determine the structure of completely finitary Gorenstein projective modules over group rings. The methods are inspired by representation theory and make use of the stable module category, in which morphisms are defined through complete cohomology. In order to carry out these methods, we need to restrict ourselves to certain classes of hierarchically decomposable groups.     

On Stability of Gorenstein Categories

We show that an iteration of the procedure used to define the Gorenstein projective modules over a ring R yields exactly the Gorenstein projective modules. Specifically, given an exact sequence of Gorenstein projective left R-modules G = … → G ₁ → G ₀ → G ⁰ → G ¹ → … such that the complex Hom _R (G, H) is exact for each projective left R-module H, the module Im(G ₀ → G ⁰) is Gorenstein projective. We also get similar results for Gorenstein flat left R-modules when R is a right coherent ring. As applications, we obtain the corresponding results for Gorenstein complexes.     

Gorenstein projective and flat complexes over noetherian rings

We give sufficient conditions on a class of R‐modules $\mathcal {C}We give sufficient conditions on a class of R‐modules $\mathcal {C}$ in order for the class of complexes of $\mathcal {C}$‐modules, $dw \mathcal {C}$, to be covering in the category of complexes of R‐modules. More precisely, we prove that if $\mathcal {C}$ is precovering in R ? Mod and if $\mathcal {C}$ is closed under direct limits, direct products, and extensions, then the class $dw \mathcal {C}$ is covering in Ch(R). Our first application concerns the class of Gorenstein flat modules. We show that when the ring R is two sided noetherian, a complex C is Gorenstein flat if and only if each module C_n is Gorenstein flat. If moreover every direct product of Gorenstein flat modules is a Gorenstein flat module, then the class of Gorenstein flat complexes is covering. We consider Gorenstein projective complexes as well. We prove that if R is a commutative noetherian ring of finite Krull dimension, then the class of Gorenstein projective complexes coincides with that of complexes of Gorenstein projective modules. We also show that if R is commutative noetherian with a dualizing complex then every right bounded complex has a Gorenstein projective precover.     

Gorenstein Injective and Gorenstein Flat Dimensions Under Base Change

Abstract

The purpose of this paper is to investigate some connections between Gorenstein flat and Gorenstein injective dimensions of complexes over different rings.     

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)     

Rings Over Which the Class of Gorenstein Flat Modules is Closed Under Extensions

A ring R is called left “GF-closed”, if the class of all Gorenstein flat left R-modules is closed under extensions. The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak dimension.

In this article, we investigate the Gorenstein flat dimension over left GF-closed rings. Namely, we generalize the fact that the class of all Gorenstein flat left modules is projectively resolving over right coherent rings to left GF-closed rings. Also, we generalize the characterization of Gorenstein flat left modules (then of Gorenstein flat dimension of left modules) over right coherent rings to left GF-closed rings. Finally, using direct products of rings, we show how to construct a left GF-closed ring that is neither right coherent nor of finite weak dimension.     

A Generalization of Auslander's Last Theorem

Before his death, Auslander announced that every finitely generated module over a local Gorenstein ring has a minimal Cohen–Macaulay approximation. Yoshimo extended Auslander's result to local Cohen–Macaulay rings admitting a dualizing module.Over a local Gorenstein ring the finitely generated maximal Cohen–Macaulay modules are the finitely generated Gorenstein projective modules so in fact Auslander's theorem says finitely generated modules over such rings have Gorenstein projective covers. We extend Auslander's theorem by proving that over a local Cohen–Macaulay ring admitting a dualizing module all finitely generated modules of finite G-dimension (in Auslander's sense) have a Gorenstein projective cover. Since all finitely generated modules over a Gorenstein ring have finite G-dimension, we recover Auslander's theorem when R is Gorenstein.     

Gorenstein AC-projective dimension of unbounded complexes

Recently, the notion of Gorenstein AC-projective (resp., Gorenstein AC-injective) modules was introduced in [3 Bravo, D., Gillespie, J., Hovey, M. The stable module category of a general ring. http://arxiv.org/abs/1405.5768. [Google Scholar]] by which the so-called “Gorenstein AC-homological algebra” was established. Here, we define and study a notion of Gorenstein AC-projective dimension for complexes (not necessarily bounded) over associative rings, which is inspired by Veliche’s construction of defining Gorenstein projective dimension. In particular, we show that such a dimension can be closely related to the “proper” Gorenstein AC-projective resolutions of complexes induced by a complete and hereditary cotorsion pair in the category of complexes of modules. This enables us to interpret this dimension of a complex in terms of vanishing of the derived functor RHom_R(?,?). As applications, some characterizations of the Gorenstein AC-projective dimension of a module are also obtained.     

Virtually Gorenstein rings and relative homology of complexes

We extend the notion of virtually Gorenstein rings to the setting of arbitrary rings, and prove that all rings R of finite Gorenstein weak global dimension are virtually Gorenstein such that all Gorenstein projective R-modules are Gorenstein flat. For such a ring R, we introduce the notion of relative homology functors of complexes with respect to Gorenstein projective (resp., flat) modules, and establish a balanced and a vanishing result for the homology functor.     

First,Second, and Third Change of Rings Theorems for Gorenstein Homological dimensions

In this article, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results established in this article for the Gorenstein projective dimension is a generalization of a G-dimension of a finitely generated module M over a noetherian ring R.     

A Hereditary Torsion Theory for Modules Over Integral Domains and Its Applications

In this article, we study the hereditary torsion theory defined by the set of associated primes of principle ideals of an integral domain, which is called the g-torsion theory. We first discuss some general properties of g-torsion theories, and after that give some applications of them. For example, we generalize a characterization of reflexive modules over quasi-normal domains to a class of non-Noetherian domains. Among other things, a characterization of coherent domains of weak Gorenstein global dimension at most two is also given in terms of Gorenstein projectivity (or Gorenstein flatness) of injective modules relative to the g-torsion theory.     

Another Gorenstein analogue of flat modules

A right R-module M is called glat if any homomorphism from any finitely presented right R-module to M factors through a finitely presented Gorenstein projective right R-module. The concept of glat modules may be viewed as another Gorenstein analogue of flat modules. We first prove that the class of glat right R-modules is closed under direct sums, direct limits, pure quotients and pure submodules for arbitrary ring R. Then we obtain that a right R-module M is glat if and only if M is a direct limit of finitely presented Gorenstein projective right R-modules. In addition, we explore the relationships between glat modules and Gorenstein flat (Gorenstein projective) modules. Finally we investigate the existence of preenvelopes and precovers by glat and finitely presented Gorenstein projective modules.