Results 1  10
of
59
Strongly Gorenstein projective, injective, and flat modules
 J. Pure Appl. Algebra
"... Abstract. In this paper, we study a particular case of Gorenstein projective, injective, and flat modules, which we call, respectively, strongly Gorenstein projective, injective, and flat modules. These last three classes of modules give us a new characterization of the first modules, and confirm th ..."
Abstract

Cited by 35 (23 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper, we study a particular case of Gorenstein projective, injective, and flat modules, which we call, respectively, strongly Gorenstein projective, injective, and flat modules. These last three classes of modules give us a new characterization of the first modules, and confirm that there is an analogy between the notion of “Gorenstein projective, injective, and flat modules”and the notion of the usual “projective, injective, and flat modules”. Key Words. Gorenstein projective, injective, and flat modules; completely projective, injective, and flat resolutions; strongly Gorenstein projective, injective, and flat modules; quasiFrobenius rings; Srings. 1
Acyclicity Versus Total Acyclicity for Complexes over Noetherian Rings
 DOCUMENTA MATH.
, 2006
"... It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is ..."
Abstract

Cited by 27 (1 self)
 Add to MetaCart
(Show Context)
It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is
Gdimension over local homomorphisms. Applications to the Frobenius endomorphism
 Ill. Jour. Math
"... Abstract. We develop a theory of Gdimension for modules over local homomorphisms which encompasses the classical theory of Gdimension for finite modules over local rings. As an application, we prove that a local ring R of characteristic p is Gorenstein if and only if it possesses a nonzero finite ..."
Abstract

Cited by 20 (10 self)
 Add to MetaCart
(Show Context)
Abstract. We develop a theory of Gdimension for modules over local homomorphisms which encompasses the classical theory of Gdimension for finite modules over local rings. As an application, we prove that a local ring R of characteristic p is Gorenstein if and only if it possesses a nonzero finite module of finite projective dimension that has finite Gdimension when considered as an Rmodule via some power of the Frobenius endomorphism of R. We also prove results that track the behavior of Gorenstein properties of local homomorphisms under (de)composition. 1.
Semidualizing modules and related Gorenstein homological dimensions
 J. Pure Appl. Algebra
"... Abstract. A semidualizing module over a commutative noetherian ring A is a finitely generated module C with RHomA(C, C) ≃ A in the derived category D(A). We show how each such module gives rise to three new homological dimensions which we call C–Gorenstein projective, C–Gorenstein injective, and C ..."
Abstract

Cited by 19 (3 self)
 Add to MetaCart
(Show Context)
Abstract. A semidualizing module over a commutative noetherian ring A is a finitely generated module C with RHomA(C, C) ≃ A in the derived category D(A). We show how each such module gives rise to three new homological dimensions which we call C–Gorenstein projective, C–Gorenstein injective, and C–Gorenstein flat dimension, and investigate the properties of these dimensions.
ASCENT PROPERTIES OF AUSLANDER CATEGORIES
, 2005
"... Let R be a homomorphic image of a Gorenstein local ring. Recent work has shown that there is a bridge between Auslander categories and modules of finite Gorenstein homological dimensions over R. We use Gorenstein dimensions to prove new results about Auslander categories and vice versa. For exampl ..."
Abstract

Cited by 13 (4 self)
 Add to MetaCart
Let R be a homomorphic image of a Gorenstein local ring. Recent work has shown that there is a bridge between Auslander categories and modules of finite Gorenstein homological dimensions over R. We use Gorenstein dimensions to prove new results about Auslander categories and vice versa. For example, we establish base change relations between the Auslander categories of the source and target rings in a homomorphism ϕ: R → S of finite flat dimension.
Foxby equivalence over associative rings
"... Abstract. We extend the definition of a semidualizing module to associative rings. This enables us to define and study Auslander and Bass classes with respect to a semidualizing bimodule C. We then study the classes of Cflats, Cprojectives, and Cinjectives, and use them to provide a characterizat ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
(Show Context)
Abstract. We extend the definition of a semidualizing module to associative rings. This enables us to define and study Auslander and Bass classes with respect to a semidualizing bimodule C. We then study the classes of Cflats, Cprojectives, and Cinjectives, and use them to provide a characterization of the modules in the Auslander and Bass classes. We extend Foxby equivalence to this new setting. This paper contains a few results which are new in the commutative, noetherian setting.
Compactly generated homotopy categories
 Homology, Homotopy Appl
"... Abstract. Over an associative ring we consider a class X of left modules which is closed under setindexed coproducts and direct summands. We investigate when the triangulated homotopy category K(X) is compactly generated, and give a number of examples. ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
Abstract. Over an associative ring we consider a class X of left modules which is closed under setindexed coproducts and direct summands. We investigate when the triangulated homotopy category K(X) is compactly generated, and give a number of examples.
THE SET OF SEMIDUALIZING COMPLEXES IS A NONTRIVIAL METRIC SPACE
, 2004
"... Abstract. There has been much speculation about the structure of the set of shiftisomorphism classes of semidualizing complexes over a local ring. In this paper we show that this set can be given the structure of a nontrivial metric space. We investigate the interplay between the metric and several ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
(Show Context)
Abstract. There has been much speculation about the structure of the set of shiftisomorphism classes of semidualizing complexes over a local ring. In this paper we show that this set can be given the structure of a nontrivial metric space. We investigate the interplay between the metric and several standard algebraic operations, and we provide a new characterization of Gorenstein rings that is motivated by this interplay. In the process, we obtain new results describing the behavior of reflexivity over homomorphisms of finite flat dimension.
A generalization of strongly Gorenstein projective modules
, 812
"... Abstract. This paper generalize the idea of the authors in J. Pure Appl. Algebra 210 (2007) 437–445. Namely, we define and study a particular case of Gorenstein projective modules. We investigate some change of rings results for this new kind of modules. Examples over not necessarily Noetherian ring ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
(Show Context)
Abstract. This paper generalize the idea of the authors in J. Pure Appl. Algebra 210 (2007) 437–445. Namely, we define and study a particular case of Gorenstein projective modules. We investigate some change of rings results for this new kind of modules. Examples over not necessarily Noetherian rings are given.
Characterizing local rings via homological dimensions and regular sequences
 J. Pure Appl. Algebra
"... Abstract. Let (R, m) be a Noetherian local ring of depth d and C a semidualizing Rcomplex. Let M be a finite Rmodule and t an integer between 0 and d. If GCdimension of M/aM is finite for all ideals a generated by an Rregular sequence of length at most d − t then either GCdimension of M is at m ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
Abstract. Let (R, m) be a Noetherian local ring of depth d and C a semidualizing Rcomplex. Let M be a finite Rmodule and t an integer between 0 and d. If GCdimension of M/aM is finite for all ideals a generated by an Rregular sequence of length at most d − t then either GCdimension of M is at most t or C is a dualizing complex. Analogous results for other homological dimensions are also given.