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97
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 42 (27 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
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 38 (2 self)
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It is proved that for a commutative noetherian ring with dualizing complex the homotopy category of projective modules is
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 32 (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.
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 31 (13 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 25 (4 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.
Global Gorenstein dimensions of polynomial rings and of direct products of rings
 Houston Journal of Mathematics Volume
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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 18 (5 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.
Global Gorenstein Dimensions
, 2007
"... Abstract. The classical global and weak dimensions of rings play an important role in the theory of rings and have a great impact on homological and commutative algebra. In this paper, we define and study the Gorenstein homological dimensions of commutative rings (Gorenstein projective, injective, a ..."
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Cited by 14 (5 self)
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Abstract. The classical global and weak dimensions of rings play an important role in the theory of rings and have a great impact on homological and commutative algebra. In this paper, we define and study the Gorenstein homological dimensions of commutative rings (Gorenstein projective, injective, and flat dimensions of rings) which introduce a new theory similar to the one of the classical homological dimensions of rings. Key Words. Classical homological dimensions of modules; global and weak dimensions of rings; Gorenstein homological dimensions of modules and of rings; strongly Gorenstein projective, injective, and flat modules; (n)Gorenstein rings; nFC rings. 1
Algebras that satisfy Auslander’s condition on vanishing of cohomology
, 2009
"... Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules. The failure of this conjecture—by a 2003 counterexample due to Jorgensen and S¸ega—motivates the consideration of the class of rings that do satisfy Auslander’s con ..."
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Cited by 14 (1 self)
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Auslander conjectured that every Artin algebra satisfies a certain condition on vanishing of cohomology of finitely generated modules. The failure of this conjecture—by a 2003 counterexample due to Jorgensen and S¸ega—motivates the consideration of the class of rings that do satisfy Auslander’s condition. We call them AC rings and show that an AC Artin algebra that is leftGorenstein is also rightGorenstein. Furthermore, the AuslanderReiten Conjecture is proved for AC rings, and Auslander’s Gdimension is shown to be functorial for AC rings that are commutative or have a dualizing complex.
Beyond totally reflexive modules and back  A survey on Gorenstein dimensions
, 2009
"... Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory’s connections with relative homological algebra and with studies of local ring homomorphisms. It ends close to the star ..."
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Cited by 11 (0 self)
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Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory’s connections with relative homological algebra and with studies of local ring homomorphisms. It ends close to the starting point: with a characterization of Gorenstein rings in terms of total acyclicity of complexes.