Related papers: Five theorems on Gorenstein global dimensions
In this paper, the $\tau_q$-weak global dimension $\tau_q$-\cwd$(R)$ of a commutative ring $R$ is introduced. Rings with $\tau_q$-weak global dimension equal to $0$ are studied in terms of homologies, direct products, polynomial extensions…
Let $R$ be a ring with Gwgldim$(R)<\infty$. We obtain a triangle-equivalence $\mathrm{K}(R\text{-}\mathrm{GProj})\simeq \mathrm{K}(R\text{-}\mathrm{GInj})$ which restricts to a triangle-equivalence $\mathrm{K}(R\text{-}\mathrm{Proj})$…
We characterize Gorenstein modules over those local rings that admit a finite contracting endomorphism.
The paper describes a duality phenomenon for cohomology theories with the character of Gorenstein rings. For a connective cohomology theory with the p-local integers in degree 0, and coefficient ring R_* Gorenstein of shift 0, this states…
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…
In this survey article we outline the history of the twin theories of weak normality and seminormality for commutative rings and algebraic varieties with an emphasis on the recent developments in these theories over the past fifteen years.…
For any group $G$, the Gorenstein homological dimension ${\rm Ghd}_RG$ is defined to be the Gorenstein flat dimension of the coefficient ring $R$, which is considered as an $RG$-module with trivial group action. We prove that ${\rm Ghd}_RG…
This paper is devoted to present some characterizations for a local ring to be generically Gorenstein and Gorenstein by means of $\delta$-invariant and linkage theory.
We prove that a finite dimensional algebra $A$ with representation-finite subcategory consisting of modules that are semi-Gorenstein-projective and $n$-th syzygy modules is left weakly Gorenstein. This generalises a theorem of Ringel and…
A notion of rigidity with respect to an arbitrary semidualizing complex C over a commutative noetherian ring R is introduced and studied. One of the main result characterizes C-rigid complexes. Specialized to the case when C is the relative…
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…
Let $G$ be a group and $R$ be a ring. We define the Gorenstein homological dimension of $G$ over $R$, denoted by ${\rm Ghd}_{R}G$, as the Gorenstein flat dimension of trivial $RG$-module $R$. It is proved that ${\rm Ghd}_SG \leq {\rm…
In this paper we study homological dimensions of finitely generated modules over commutative Noetherian local rings, called reducing homological dimensions. We obtain new characterizations of Gorenstein and complete intersection local rings…
Motivated by their impact on homological algebra, the change of rings results have been the subject of several interesting works in Gorenstein homological algebra over Noetherian rings. In this paper, we investigate the change of rings…
In \cite{Ouarghi}, the authors discuss the rings over which all modules are strongly Gorenstein projective. In this paper, we are interesting to an extension of this idea. Thus, we discuss the rings over which every Gorenstein projective…
Weakly locally finite division rings were considered in \cite{dbh}, where it was mentioned that the class of weakly locally finite division rings properly contains the class of locally finite division rings. In this paper, for any integer…
The strong global dimension of a ring is the supremum of the length of perfect complexes that are indecomposable in the derived category. In this note we characterize the noetherian commutative rings that have finite strong global…
In 1969, Osofsky proved that a chained ring (i.e., local arithmetical ring) with zero divisors has infinite weak global dimension; that is, the weak global dimension of an arithmetical ring is 0, 1, or infinite. In 2007, Bazzoni and Glaz…
The notion of almost Gorenstein local ring introduced by V. Barucci and R. Fr\"oberg for one-dimensional Noetherian local rings which are analytically unramified has been generalized by S. Goto, N. Matsuoka and T. T. Phuong to…
The notion of generalized Gorenstein local ring (GGL ring for short) is one of the generalizations of Gorenstein rings. In this article, there is given a characterization of GGL rings in terms of their canonical ideals and related…