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47
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 ..."
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Cited by 30 (22 self)
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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
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 ..."
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Cited by 18 (4 self)
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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.
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 ..."
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Cited by 18 (1 self)
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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 ..."
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Cited by 17 (9 self)
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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.
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 ..."
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Cited by 11 (5 self)
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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.
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. ..."
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Cited by 8 (1 self)
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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.
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 ..."
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Cited by 8 (4 self)
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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.
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 ..."
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Cited by 7 (3 self)
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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.
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 ..."
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Cited by 5 (4 self)
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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.
GORENSTEIN HOMOLOGICAL DIMENSIONS AND AUSLANDER CATEGORIES
, 2006
"... Abstract. In this paper, we study Gorenstein injective, projective, and flat modules over a Noetherian ring R. For an Rmodule M, we denote by GpdRM and GfdRM the Gorenstein projective and flat dimensions of M, respectively. We show that GpdRM < ∞ if and only if GfdRM < ∞ provided the Krull dimensio ..."
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Cited by 4 (0 self)
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Abstract. In this paper, we study Gorenstein injective, projective, and flat modules over a Noetherian ring R. For an Rmodule M, we denote by GpdRM and GfdRM the Gorenstein projective and flat dimensions of M, respectively. We show that GpdRM < ∞ if and only if GfdRM < ∞ provided the Krull dimension of R is finite. Moreover, in the case that R is local, we correspond to a dualizing complex D of ˆ R, the classes A ′(R) and B ′(R) of Rmodules. For a module M over a local ring R, we show that M ∈ A ′(R) if and only if GpdRM < ∞ or equivalently GfdRM < ∞. In dual situation by using the class B ′(R), we provide a characterization of Gorenstein injective modules. 1.