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38
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.
Global Hochschild (co)homology of singular spaces
, 2006
"... We introduce Hochschild (co)homology of morphisms of schemes or analytic spaces and study its fundamental properties. In analogy with the cotangent complex we introduce the so called (derived) Hochschild complex of a morphism; the Hochschild cohomology and homology groups are then the Ext and Tor ..."
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Cited by 19 (3 self)
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We introduce Hochschild (co)homology of morphisms of schemes or analytic spaces and study its fundamental properties. In analogy with the cotangent complex we introduce the so called (derived) Hochschild complex of a morphism; the Hochschild cohomology and homology groups are then the Ext and Tor groups of that complex. We prove that these objects are well defined, extend the known cases, and have the expected functorial and homological properties such as graded commutativity of Hochschild cohomology and existence of the characteristic homomorphism from Hochschild cohomology to the (graded) centre of the derived category.
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
Multiplicative structures for Koszul algebras
, 2005
"... Abstract. Let Λ = kQ/I be a Koszul algebra over a field k, where Q is a finite quiver. An algorithmic method for finding a minimal projective resolution F of the graded simple modules over Λ is given in [9]. This resolution is shown to have a “comultiplicative ” structure in [7], and this is used to ..."
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Cited by 17 (6 self)
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Abstract. Let Λ = kQ/I be a Koszul algebra over a field k, where Q is a finite quiver. An algorithmic method for finding a minimal projective resolution F of the graded simple modules over Λ is given in [9]. This resolution is shown to have a “comultiplicative ” structure in [7], and this is used to find a minimal projective resolution P of Λ over the enveloping algebra Λ e. Using these results we show that the multiplication in the Hochschild cohomology ring of Λ relative to the resolution P is given as a cup product and also provide a description of this product. This comultiplicative structure also yields the structure constants of the Koszul dual of Λ with respect to a canonical basis over k associated to the resolution F. The natural map from the Hochschild cohomology to the Koszul dual of Λ is shown to be surjective onto the graded centre of the Koszul dual.
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.
Maximal northogonal modules for selfinjective algebras
 PROC. AMER. MATH. SOC
, 2008
"... Let A be a finitedimensional selfinjective algebra. We show that, for any n ≥ 1, maximal northogonal Amodules (in the sense of Iyama) rarely exist. More precisely, we prove that if A admits a maximal northogonal module, then all Amodules are of complexity at most 1. ..."
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Cited by 13 (0 self)
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Let A be a finitedimensional selfinjective algebra. We show that, for any n ≥ 1, maximal northogonal Amodules (in the sense of Iyama) rarely exist. More precisely, we prove that if A admits a maximal northogonal module, then all Amodules are of complexity at most 1.
Homology and cohomology of quantum complete intersections
 Algebra Number Theory
"... Dedicated to Lucho Avramov on the occasion of his sixtieth birthday ..."
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Cited by 11 (2 self)
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Dedicated to Lucho Avramov on the occasion of his sixtieth birthday
Modules with reducible complexity
 J. Algebra
"... Abstract. For a commutative Noetherian local ring we define and study the class of modules having reducible complexity, a class containing all modules of finite complete intersection dimension. Various properties of this class of modules are given, together with results on the vanishing of homology ..."
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Cited by 10 (6 self)
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Abstract. For a commutative Noetherian local ring we define and study the class of modules having reducible complexity, a class containing all modules of finite complete intersection dimension. Various properties of this class of modules are given, together with results on the vanishing of homology and cohomology. 1.
On support varieties of modules over complete intersections
 Proc. Amer. Math. Soc
"... Abstract. Let (A, m, k) be a complete intersection of codimension c, and ˜ k the algebraic closure of k. We show that every homogeneous algebraic subset of ˜k c is the cohomological support variety of an Amodule, and that the projective variety of a complete indecomposable maximal CohenMacaulay A ..."
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Cited by 9 (3 self)
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Abstract. Let (A, m, k) be a complete intersection of codimension c, and ˜ k the algebraic closure of k. We show that every homogeneous algebraic subset of ˜k c is the cohomological support variety of an Amodule, and that the projective variety of a complete indecomposable maximal CohenMacaulay Amodule is connected. 1.