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26
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
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
Constructions for infinitesimal group schemes
, 2010
"... Let G be an infinitesimal group scheme over a field k of characteristic p> 0. We introduce the global pnilpotent operator ΘG: k[G] → k[V (G)], where V (G) is the scheme which represents 1parameter subgroups of G. This operator ΘG applied to M encodes the local Jordan type of M, and leads to c ..."
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Cited by 12 (8 self)
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Let G be an infinitesimal group scheme over a field k of characteristic p> 0. We introduce the global pnilpotent operator ΘG: k[G] → k[V (G)], where V (G) is the scheme which represents 1parameter subgroups of G. This operator ΘG applied to M encodes the local Jordan type of M, and leads to computational insights into the representation theory of G. For certain kGmodules M (including those of constant Jordan type), we employ ΘG to associate various algebraic vector bundles on P(G), the projectivization of V (G). These vector bundles not only distinguish certain representations with the same local Jordan type, but also provide a method of constructing algebraic vector bundles on P(G).
GENERALIZED SUPPORT VARIETIES FOR FINITE GROUP SCHEMES
, 2010
"... We construct two families of refinements of the (projectivized) support variety of a finite dimensional module M for a finite group scheme G. For an arbitrary finite group scheme, we associate a family of non maximal subvarieties Γ(G) j M, 1 ≤ j ≤ p−1, to a kGmodule M. For G infinitesimal, we cons ..."
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Cited by 8 (4 self)
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We construct two families of refinements of the (projectivized) support variety of a finite dimensional module M for a finite group scheme G. For an arbitrary finite group scheme, we associate a family of non maximal subvarieties Γ(G) j M, 1 ≤ j ≤ p−1, to a kGmodule M. For G infinitesimal, we construct a finer family of locally closed subvarieties Γa (G)M for any partition a of dim M. We give a cohomological interpretation of the varieties Γ1 (G)M for certain modules relating them to generalizations of Z(ζ), the zero loci of cohomology classes ζ ∈ H • (G, k).
Generalized trace and modified dimension functions on ribbon categories
 Selecta Math., Volume 17, Issue
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STACKS OF GROUP REPRESENTATIONS
"... Abstract. We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extensionofscalars. We deduce that, given a group G, the derived and the stable categories of representations of a subgroup H can be constructed ..."
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Cited by 7 (4 self)
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Abstract. We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extensionofscalars. We deduce that, given a group G, the derived and the stable categories of representations of a subgroup H can be constructed out of the corresponding category for G by a purely triangulatedcategorical construction, analogous to étale extension in algebraic geometry. In the case of finite groups, we then use descent methods to investigate when modular representations of the subgroup H can be extended to G. We show that the presheaves of plain, derived and stable representations all form
On injective modules and support varieties for the small quantum group
 International Mathematics Research Notices
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REPRESENTATIONS OF ELEMENTARY ABELIAN pGROUPS AND BUNDLES ON GRASSMANNIANS
, 2011
"... We initiate the study of representations of elementary abelian pgroups via restrictions to truncated polynomial subalgebras of the group algebra generated by r nilpotent elements, k[t1,..., tr]/(t p 1,..., tpr). We introduce new geometric invariants based on the behavior of modules upon restrictio ..."
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Cited by 3 (1 self)
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We initiate the study of representations of elementary abelian pgroups via restrictions to truncated polynomial subalgebras of the group algebra generated by r nilpotent elements, k[t1,..., tr]/(t p 1,..., tpr). We introduce new geometric invariants based on the behavior of modules upon restrictions to such subalgebras. We also introduce modules of constant radical and socle type generalizing modules of constant Jordan type and provide several general constructions of modules with these properties. We show that modules of constant radical and socle type lead to families of algebraic vector bundles on