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Πsupports for modules for finite group schemes
"... Abstract. We introduce the space Π(G) of equivalence classes of πpoints of a finite group scheme G, and associate a subspace Π(G)M to any Gmodule M. Our results extend to arbitrary finite group schemes G over arbitrary fields k of positive characteristic and to arbitrarily large Gmodules the basi ..."
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Cited by 26 (8 self)
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Abstract. We introduce the space Π(G) of equivalence classes of πpoints of a finite group scheme G, and associate a subspace Π(G)M to any Gmodule M. Our results extend to arbitrary finite group schemes G over arbitrary fields k of positive characteristic and to arbitrarily large Gmodules the basic results about “cohomological support varieties ” and their interpretation in terms of representation theory. In particular, we prove that the projectivity of any (possibly infinite dimensional) Gmodule can be detected by its restriction along πpoints of G. Unlike the cohomological support variety of a Gmodule M, the invariant M ↦ → Π(G)M satisfies good properties for all modules, thereby enabling us to determine the thick, tensorideal subcategories of the stable
COHOMOLOGY AND SUPPORT VARIETIES FOR LIE SUPERALGEBRAS
, 2006
"... 1.1. The blocks of the Category O (or relative Category OS) for complex semisimple Lie algebras are well known examples of highest weight categories, as defined in [CPS], with finitely many simple modules. These facts imply that the projective resolutions for modules in these categories have finite ..."
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Cited by 18 (4 self)
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1.1. The blocks of the Category O (or relative Category OS) for complex semisimple Lie algebras are well known examples of highest weight categories, as defined in [CPS], with finitely many simple modules. These facts imply that the projective resolutions for modules in these categories have finite length, so the cohomology (or extensions) can be nonzero in
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.
Support varieties and representation type of small quantum groups
 Internat. Math. Res. Notices
"... Abstract. In this paper we provide a wildness criterion for any finite dimensional Hopf algebra with finitely generated cohomology. This generalizes a result of Farnsteiner to not necessarily cocommutative Hopf algebras over ground fields of arbitrary characteristic. Our proof uses the theory of su ..."
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Cited by 14 (6 self)
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Abstract. In this paper we provide a wildness criterion for any finite dimensional Hopf algebra with finitely generated cohomology. This generalizes a result of Farnsteiner to not necessarily cocommutative Hopf algebras over ground fields of arbitrary characteristic. Our proof uses the theory of support varieties for modules, one of the crucial ingredients being a tensor product property for some special modules. As an application we prove a conjecture of Cibils stating that small quantum groups of rank at least two are wild. 1.
Generic and Maximal Jordan types
"... Abstract. For a finite group scheme G over a field k of characteristic p> 0, we associate new invariants to a finite dimensional kGmodule M. Namely, for each generic point of the projectivized cohomological variety Proj H • (G, k) we exhibit a “generic Jordan type ” of M. In the very special cas ..."
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Cited by 14 (3 self)
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Abstract. For a finite group scheme G over a field k of characteristic p> 0, we associate new invariants to a finite dimensional kGmodule M. Namely, for each generic point of the projectivized cohomological variety Proj H • (G, k) we exhibit a “generic Jordan type ” of M. In the very special case in which G = E is an elementary abelian pgroup, our construction specializes to the nontrivial observation that the Jordan type obtained by restricting M via a generic cyclic shifted subgroup does not depend upon a choice of generators for E. Furthermore, we construct the nonmaximal support variety Γ(G)M, a closed subset of Proj H • (G, k) which is proper even when the dimension of M is not divisible by p. Elementary abelian psubgroups of a finite group G capture significant aspects of the cohomology and representation theory of G. For example, if k is a field of characteristic p> 0, then a theorem of D. Quillen [18] asserts that the Krull dimension of the cohomology algebra H • (G, k) is equal to the maximum of the ranks of
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).
Tameness and complexity of finite group schemes
 Bull. London Math. Soc
"... Abstract. Using a representationtheoretic interpretation of support varieties due to FriedlanderPevtsova [18], we show that the complexity of tame blocks of finite group schemes is bounded by 2. In this context, our result salvages a theorem by Rickard [23], whose proof is flawed. ..."
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Cited by 11 (0 self)
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Abstract. Using a representationtheoretic interpretation of support varieties due to FriedlanderPevtsova [18], we show that the complexity of tame blocks of finite group schemes is bounded by 2. In this context, our result salvages a theorem by Rickard [23], whose proof is flawed.
Complexity and module varieties for classical Lie superalgebras; arXiv:0905.2403
"... Abstract. Let g = g¯0 ⊕g¯1 be a classical Lie superalgebra and F be the category of finite dimensional gsupermodules which are semisimple over g¯0. In this paper we investigate the homological properties of the category F. In particular we prove that F is selfinjective in the sense that all project ..."
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Abstract. Let g = g¯0 ⊕g¯1 be a classical Lie superalgebra and F be the category of finite dimensional gsupermodules which are semisimple over g¯0. In this paper we investigate the homological properties of the category F. In particular we prove that F is selfinjective in the sense that all projective supermodules are injective. We also show that all supermodules in F admit a projective resolution with polynomial rate of growth and, hence, one can study complexity in F. If g is a Type I Lie superalgebra we introduce support varieties which detect projectivity and are related to the associated varieties of Duflo and Serganova. If in addition g has a (strong) duality then we prove that the conditions of being tilting or projective are equivalent. 1.
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).
SUPPORT VARIETIES FOR FROBENIUS KERNELS OF CLASSICAL GROUPS
"... Abstract. Let G be a classical simple algebraic group over an algebraically closed field k of characteristic p> 0, and denote by G (r) the rth Frobenius kernel of G. We show that for p large enough, the support variety of a simple Gmodule over G (r) can be described in terms of support varietie ..."
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Cited by 6 (2 self)
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Abstract. Let G be a classical simple algebraic group over an algebraically closed field k of characteristic p> 0, and denote by G (r) the rth Frobenius kernel of G. We show that for p large enough, the support variety of a simple Gmodule over G (r) can be described in terms of support varieties of simple Gmodules over G (1). We use this, together with the computation of the varieties VG (1) (H 0 (λ)), given by Jantzen in [8] and by Nakano et al. in [10], to explicitly compute the support variety of a block of Dist(G (r)). The aim of this paper is to provide computations of support varieties for modules over Frobenius kernels of algebraic groups. Specifically, for G a classical simple algebraic group over an algebraically closed field k of characteristic p> 0, we give a description (Theorem 3.2) of the support variety of a simple Gmodule over the rth Frobenius kernel G (r) in terms of the support varieties of simple Gmodules over G (1). Our proofs establish these results only under the assumption that p is large enough for the root system of G. A lower bound on p is provided in Section 3,