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77
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.
Support varieties for selfinjective algebras
 KTheory
"... Abstract. Support varieties for any finite dimensional algebra over a field were introduced in [20] using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from ..."
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Cited by 38 (12 self)
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Abstract. Support varieties for any finite dimensional algebra over a field were introduced in [20] using graded subalgebras of the Hochschild cohomology. We mainly study these varieties for selfinjective algebras under appropriate finite generation hypotheses. Then many of the standard results from the theory of support varieties for finite groups generalize to this situation. In particular, the complexity of the module equals the dimension of its corresponding variety, all closed homogeneous varieties occur as the variety of some module, the variety of an indecomposable module is connected, periodic modules are lines and for symmetric algebras a generalization of Webb’s theorem is true.
Classifying thick subcategories of the stable category of CohenMacaulay modules
, 2009
"... Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a fin ..."
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Cited by 23 (7 self)
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Various classification theorems of thick subcategories of a triangulated category have been obtained in many areas of mathematics. In this paper, as a higher dimensional version of the classification theorem of thick subcategories of the stable category of finitely generated representations of a finite pgroup due to Benson, Carlson and Rickard, we consider classifying thick subcategories of the stable category of CohenMacaulay modules over a Gorenstein local ring. The main result of this paper yields a complete classification of the thick subcategories of the stable category of CohenMacaulay modules over a local hypersurface in terms of specializationclosed subsets of the prime ideal spectrum of the ring which are contained in its singular locus. We also consider classifying resolving subcategories of the category of finitely generated modules. Our method also gives some information on the structure of CohenMacaulay modules
Independence of the total reflexivity conditions for modules
"... Abstract. We show that the conditions defining total reflexivity for modules are independent. In particular, we construct a commutative Noetherian local ring R and a reflexive Rmodule M such that Exti R (M, R) = 0 for all i> 0, but Exti R (M ∗ , R) ̸ = 0 for all i> 0. introduction Let R be a ..."
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Cited by 18 (2 self)
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Abstract. We show that the conditions defining total reflexivity for modules are independent. In particular, we construct a commutative Noetherian local ring R and a reflexive Rmodule M such that Exti R (M, R) = 0 for all i> 0, but Exti R (M ∗ , R) ̸ = 0 for all i> 0. introduction Let R be a commutative Noetherian ring. For any Rmodule M we set M ∗ = HomR(M, R). An Rmodule M is said to be reflexive if it is finite and the canonical map M → M ∗ ∗ is bijective. A finite Rmodule M is said to be totally reflexive if it satisfies the following conditions: (i) M is reflexive (M, R) = 0 for all i> 0 (ii) Ext i R (iii) Ext i R (M ∗ , R) = 0 for all i> 0. This notion is due to Auslander and Bridger [1]: the totally reflexive modules are precisely the modules of Gdimension zero. The Gdimension of a module is one of the best studied nonclassical homological dimensions, and is defined in terms of the length of a resolution of the module by totally reflexive modules. Given any homological dimension, a serious concern is whether its defining conditions can be verified effectively. For example, the projective dimension of a finite Rmodule M is zero if and only if Ext 1 R(M, N) = 0 for all finite Rmodules N. However, when R is local with maximal ideal m, one only needs to check vanishing for N = R/m. In the same spirit, it is natural to ask whether the set of conditions defining total reflexivity is overdetermined (cf. [4, §2]) and in particular, whether total reflexivity for a module can be established by verifying vanishing of only finitely many Ext modules. When R is a local Gorenstein ring, (ii) implies the other two conditions above, and it is equivalent to M being maximal CohenMacaulay. Recently, Yoshino [9] studied other situations when (ii) alone implies total reflexivity, and raised the question whether this is always the case. In the present paper we give an example of a local Artinian ring R which admits modules whose total reflexivity conditions are independent, in that (ii) implies
Stratifying triangulated categories
, 2009
"... A notion of stratification is introduced for any compactly generated triangulated category T endowed with an action of a graded commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences which follow when T is stratified by R. Among them are a cl ..."
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Cited by 18 (5 self)
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A notion of stratification is introduced for any compactly generated triangulated category T endowed with an action of a graded commutative noetherian ring R. The utility of this notion is demonstrated by establishing diverse consequences which follow when T is stratified by R. Among them are a classification of the localizing subcategories of T in terms of subsets of the set of prime ideals in R; a classification of the thick subcategories of the subcategory of compact objects in T; and results concerning the support of the Rmodule of homomorphisms Hom ∗ T (C, D) leading to an analogue of the tensor
Vanishing Theorems For Complete Intersections
, 2000
"... this paper deals with trying to prove the modules must be reflexive in case the tensor product is reflexive. The following easy inductive argument will be used in the proofs of both of the main theorems (2.4) and (2.7). ..."
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Cited by 17 (2 self)
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this paper deals with trying to prove the modules must be reflexive in case the tensor product is reflexive. The following easy inductive argument will be used in the proofs of both of the main theorems (2.4) and (2.7).
VARIETIES FOR MODULES OF QUANTUM ELEMENTARY ABELIAN GROUPS
"... Abstract. We define a rank variety for a module of a noncocommutative Hopf algebra A = Λ ⋊ G where Λ = k[X1,..., Xm]/(X ℓ 1,..., X ℓ m), G = (Z/ℓZ) m, and char k does not divide ℓ, in terms of certain subalgebras of A playing the role of “cyclic shifted subgroups”. We show that the rank variety of a ..."
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Cited by 17 (6 self)
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Abstract. We define a rank variety for a module of a noncocommutative Hopf algebra A = Λ ⋊ G where Λ = k[X1,..., Xm]/(X ℓ 1,..., X ℓ m), G = (Z/ℓZ) m, and char k does not divide ℓ, in terms of certain subalgebras of A playing the role of “cyclic shifted subgroups”. We show that the rank variety of a finitely generated module M is homeomorphic to the support variety of M defined in terms of the action of the cohomology algebra of A. As an application we derive a theory of rank varieties for the algebra Λ. When ℓ = 2, rank varieties for Λmodules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for Λmodules coincide with those of Erdmann and Holloway. 1.