Note on (weak) Gorenstein global dimensions

Journal of Algebra and Its Applications, 2016

Given a semidualizing module [Formula: see text] over a commutative Noetherian ring, Holm and Jørgensen [Semi-dualizing modules and related Gorenstein homological dimensions, J. Pure Appl. Algebra 205(2) (2006) 423–445] investigate some connections between [Formula: see text]-Gorenstein dimensions of an [Formula: see text]-complex and Gorenstein dimensions of the same complex viewed as a complex over the “trivial extension” [Formula: see text]. We generalize some of their results to a certain type of retract diagram. We also investigate some examples of such retract diagrams, namely D’Anna and Fontana’s amalgamated duplication [An amalgamated duplication of a ring along an ideal: The basic properties, J. Algebra Appl. 6(3) (2007) 443–459] and Enescu’s pseudocanonical cover [A finiteness condition on local cohomology in positive characteristic, J. Pure Appl. Algebra 216(1) (2012) 115–118].

Proceedings of the American Mathematical Society, 2009

In this paper, we prove that the global Gorenstein projective dimension of a ring R is equal to the global Gorenstein injective dimension of R, and that the global Gorenstein flat dimension of R is smaller than the common value of the terms of this equality.

Journal of Pure and Applied Algebra, 2004

In basic homological algebra, the projective, injective and at dimensions of modules play an important and fundamental role. In this paper, the closely related Gorenstein projective, Gorenstein injective and Gorenstein at dimensions are studied.

Electronic Research Archive, 2020

Let G(X) and G(Y) be Gorenstein subcategories induced by an admissible balanced pair (X , Y) in an abelian category A. In this paper, we establish Gorenstein homological dimensions in terms of these two subcategories and investigate the Gorenstein global dimensions of A induced by the balanced pair (X , Y). As a consequence, we give some new characterizations of pure global dimensions and Gorenstein global dimensions of a ring R.

Chinese Annals of Mathematics, Series B, 2011

There is a variety of nice results about strongly Gorenstein flat modules over coherent rings. These results are done by Ding, Lie and Mao. The aim of this paper is to generalize some of these results, and to give homological descriptions of the strongly Gorenstein flat dimension (of modules and rings) over arbitrary associative rings. 534 N. Mahdou and M. Tamekkante

We compute the Gorenstein weak dimension of a coherent power series rings over a commutative rings and we show that, in general, $\gwd(R) \leq 1$ does not imply that $R$ is an arithmetical ring.

Mediterranean Journal of Mathematics, 2009

In this paper, we study the Gorenstein global dimension of an amalgamated duplication of a coherent ring along a regular principal ideal.

2020

It is a well-known result of Auslander and Reiten that contravariant finiteness of the class P^fin_∞ (of finitely generated modules of finite projective dimension) over an Artin algebra is a sufficient condition for validity of finitistic dimension conjectures. Motivated by the fact that finitistic dimensions of an algebra can alternatively be computed by Gorenstein projective dimension, in this work we examine the Gorenstein counterpart of Auslander–Reiten condition, namely contravariant finiteness of the class GP^fin_∞ (of finitely generated modules of finite Gorenstein projective dimension), and its relation to validity of finitistic dimension conjectures. It is proved that contravariant finiteness of the class GP^fin_∞ implies validity of the second finitistic dimension conjecture over left artinian rings. In the more special setting of Artin algebras, however, it is proved that the Auslander–Reiten sufficient condition and its Gorenstein counterpart are virtually equivalent in ...

Communications in Algebra, 2020

Let u : R ! S be a local homomorphism and X a homologically finite S-complex. Iyengar and Sather-Wagstaff defined the G-dimension and projective dimension of X over u by Cohen factorizations and Avramov, Iyengar, and Miller introduced the injective dimension of X over u: In this article, we define the Gorenstein injective dimension of X over u: The Bass equality for the Gorenstein injective dimension over local homomorphism is obtained, which generalizes a result of Christensen and Sather-Wagstaff. Some other known results are generalized. For instance, if id u X is finite, then it is shown that G-dim u X ¼ pd u X: This is analogous to a theorem of Holm.

2008

In this paper, we introduce and study the rings of Gorenstein homological dimensions small or equal than 1, which we call Gorenstein (semi)hereditary rings, specially particular cases of these rings, which we call strongly Gorenstein (semi)hereditary rings.