Rings with finite Gorenstein injective dimension

In this paper we prove that for any associative ring , and for any left -module with finite projective dimension, the Gorenstein injective dimension equals the usual injective dimension . In particular, if is finite, then also is finite, and thus is Gorenstein (provided that is commutative and Noetherian).

     

Gorenstein dimension of modules over homomorphisms

Given a homomorphism of commutative noetherian rings R→S and an S-module N, it is proved that the Gorenstein flat dimension of N over R, when finite, may be computed locally over S. When, in addition, the homomorphism is local and N is finitely generated over S, the Gorenstein flat dimension equals , where E is the injective hull of the residue field of R. This result is analogous to a theorem of André on flat dimension.     

A Bass equality for Gorenstein injective dimension of modules finite over homomorphisms

Archiv der Mathematik - The groups having exactly one normalizer are Dedekind groups. All finite groups with exactly two normalizers were classified by Pérez-Ramos in 1988. In this paper we...     

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 3 1i \geq 1 Extⁱ (E,M) = 0 for all indecomposable injective R-modules E 1 E(k)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 3 dn \geq d.     

Selforthogonal modules with finite injective dimension

The category consisting of finitely generated modules which are left orthogonal with a cotilting bimodule is shown to be functorially finite. The notion of left orthogonal dimension is introduced, and then a necessary and sufficient condition of selforthogonal modules having finite injective dimension and a characterization of cotilting modules are given.     

On the category of modules of Gorenstein dimension zero

Let R be a commutative noetherian henselian non-Gorenstein local ring of depth zero. Denote by modR the category of all finitely generated R-modules, and by the full subcategory of modR consisting of all R-modules of Gorenstein dimension zero. We prove in this paper that if contains a non-free module, then it is not precovering in modR, in particular, there exist infinitely many isomorphism classes of indecomposable R-modules of Gorenstein dimension zero.     

Wakamatsu tilting modules with finite injective dimension

Let R be a left Noetherian ring, S a right Noetherian ring and _R ω a Wakamatsu tilting module with S = End( _R ω). We introduce the notion of the ω-torsionfree dimension of finitely generated R-modules and give some criteria for computing it. For any n ? 0, we prove that l.id _R (ω) = r.id _S (ω) ? n if and only if every finitely generated left R-module and every finitely generated right S-module have ω-torsionfree dimension at most n, if and only if every finitely generated left R-module (or right S-module) has generalized Gorenstein dimension at most n. Then some examples and applications are given.     

Gorenstein fp-平坦和强Gorenstein fp-平坦模

引入了Gorenstein fp-平坦模和强Gorenstein fp-平坦模的概念,讨论了这两类模的一些性质、联系以及稳定性.     

Strongly Gorenstein graded modules

Let R be a graded ring. We define and study strongly Gorenstein gr-projective, gr-injective, and gr-flat modules. Some connections among these modules are discussed. We also explore the relations between the graded and the ungraded strongly Gorenstein modules.     

n- 强Gorenstein FP- 内射模

设n 是正整数, 本文引入并研究n- 强Gorenstein FP- 内射模. 对于正整数n > m, 给出例子说明n- 强Gorenstein FP- 内射模未必是m- 强Gorenstein FP- 内射的, 并讨论n- 强Gorenstein FP-内射模的诸多性质. 最后, 利用n- 强Gorenstein FP- 内射模刻画n- 强Gorenstein Von Neumann 正则环.     

A characterization of modules locally of finite injective dimension

In this note, we characterize finite modules locally of finite injective dimension over commutative Noetherian rings in terms of vanishing of Ext modules.

     

Contracting endomorphisms and Gorenstein modules

A finite module M over a noetherian local ring R is said to be Gorenstein if Extⁱ(k, M) = 0 for all i ≠ dim R. An endomorphism φ: R → R of rings is called contracting if for some i ≥ 1. Letting ^φR denote the R-module R with action induced by φ, we prove: A finite R-module M is Gorenstein if and only if Hom_R(^φR, M) ≅ M and Extⁱ_R(^φR, M) = 0 for 1 ≤ i ≤ depth R. Received: 7 December 2007     

On modules of finite projective dimension over complete intersections

Recently Avramov and Miller proved that over a local complete intersection ring in characteristic 0$">, a finitely generated module has finite projective dimension if for some 0$"> and for some 0$">, being the frobenius map repeated times. They used the notion of ``complexity' and several related theorems. Here we offer a very simple proof of the above theorem without using ``complexity' at all.

     

Gorenstein injective, Gorenstein flat modules and the section functor

Let R be a commutative Noetherian ring of Krull dimension d, and let a be an ideal of R. In this paper, we will study the strong cotorsioness and the Gorenstein injectivity of the section functor Γ_a(−) in local cohomology. As applications, we will find new characterizations for Gorenstein and regular local rings. We also study the effect of the section functors Γ_a(−) and the functors on the Auslander and Bass classes.