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.     

Remarks on torsionfreeness and its applications

In this article, we shall characterize torsionfreeness of modules with respect to a semidualizing module in terms of the Serre’s condition (S_n). As its applications, we give a characterization of Cohen-Macaulay rings R such that R_𝔭 is Gorenstein for all prime ideals 𝔭 of height less than n, and we will give a partial answer of Tachikawa conjecture and Auslander-Reiten conjecture.     

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 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.     

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.     

模的强Gorenstein平坦维数

In the Gorenstein homological theory, Gorenstein projective and Gorenstein injective dimensions play an important and fundamental role. In this paper, we aim at studying the closely related strongly Gorenstein flat and Gorenstein FP-injective dimensions, and show that some characterizations similar to Gorenstein homological dimensions hold for these two dimensions.     

Characterizing Local Rings via Test Modules

This article shows some instances where properties of a local ring are closely connected with the homological properties of a single module.     

Gorenstein projective dimensions of complexes

We show that over a right coherent left perfect ring R, a complex C of left R-modules is Gorenstein projective if and only if C ^m is Gorenstein projective in R-Mod for all m ∈ ℤ. Basing on this we show that if R is a right coherent left perfect ring then Gpd(C) = sup{Gpd(C ^m )|m ∈ ℤ} where Gpd(−) denotes Gorenstein projective dimension.     

Auslander–Reiten Triangles in Homotopy Categories

Let A be an artin algebra. We show that the bounded homotopy category of finitely generated right A-modules has Auslander–Reiten triangles. Two applications are given: (1) we provide an alternative proof of a theorem of Happel in [14 Happel, D. (1988). Triangulated Categories in the Representation Theory of Finite-dimensional Algebras. London Mathematical Society Lecture Note Series, vol. 119. Cambridge: Cambridge University Press.[Crossref] , [Google Scholar]]; (2) we prove that over a Gorenstein algebra, the bounded homotopy category of finitely generated Gorenstein projective (resp. injective) modules, admits Auslander–Reiten triangles, which improve a main result in [12 Nan, G. (2012). Auslander-Reiten triangles on Gorenstein derived categories. Comm. Algebra 40:3912–3919.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].