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Local cohomology and support for triangulated categories
, 2007
"... We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated categories, with respect to a central ring of operators. Suitably ..."
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Cited by 58 (19 self)
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We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated categories, with respect to a central ring of operators. Suitably specialized one recovers, for example, the theory for commutative noetherian rings due to Foxby and Neeman, the theory of Avramov and Buchweitz for complete intersection local rings, and varieties for representations of finite groups according to Benson, Carlson, and Rickard. We give explicit examples of objects whose triangulated support and cohomological support differ. In the case of group representations, this leads to a counterexample to a conjecture of Benson.
DIMENSIONS OF TRIANGULATED CATEGORIES VIA KOSZUL OBJECTS
"... Abstract. Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin alge ..."
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Cited by 20 (10 self)
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Abstract. Lower bounds for the dimension of a triangulated category are provided. These bounds are applied to stable derived categories of Artin algebras and of commutative complete intersection local rings. As a consequence, one obtains bounds for the representation dimensions of certain Artin algebras. 1.
Finite Gorenstein representation type implies simple singularity
, 2008
"... Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive Rmodules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity ..."
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Cited by 17 (6 self)
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Let R be a commutative noetherian local ring and consider the set of isomorphism classes of indecomposable totally reflexive Rmodules. We prove that if this set is finite, then either it has exactly one element, represented by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R is complete, then it is even a simple hypersurface singularity). The crux of our proof is to argue that if the residue field has a totally reflexive cover, then R is Gorenstein or every totally reflexive Rmodule is free.
COHOMOLOGICAL SYMMETRY IN TRIANGULATED CATEGORIES
"... Abstract. We give a criterion for cohomological symmetry in a triangulated ..."
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Cited by 2 (0 self)
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Abstract. We give a criterion for cohomological symmetry in a triangulated
ON GROWTH IN TOTALLY ACYCLIC MINIMAL COMPLEXES
, 904
"... Abstract. Given a commutative Noetherian local ring, we provide a criterion under which a totally acyclic minimal complex of free modules has symmetric growth. 1. ..."
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Cited by 1 (0 self)
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Abstract. Given a commutative Noetherian local ring, we provide a criterion under which a totally acyclic minimal complex of free modules has symmetric growth. 1.
ON GROWTH IN MINIMAL TOTALLY ACYCLIC COMPLEXES
"... Abstract. Given a commutative Noetherian local ring, we provide a criterion under which a minimal totally acyclic complex of free modules has symmetric growth. As a special case, we show that whenever an image in the complex has finite complete intersection dimension, then the complex has symmetric ..."
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Abstract. Given a commutative Noetherian local ring, we provide a criterion under which a minimal totally acyclic complex of free modules has symmetric growth. As a special case, we show that whenever an image in the complex has finite complete intersection dimension, then the complex has symmetric polynomial growth. 1.