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23
COHOMOLOGY OF FINITE GROUP SCHEMES OVER A FIELD
"... A finite group scheme G over a field k is equivalent to its coordinate algebra, a finite dimensional commutative Hopf algebra k[G] over k. In many contexts, it is natural to consider the rational (or Hochschild) cohomology of G with coefficients in a k[G]comodule M. This is naturally isomorphic to ..."
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Cited by 54 (10 self)
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A finite group scheme G over a field k is equivalent to its coordinate algebra, a finite dimensional commutative Hopf algebra k[G] over k. In many contexts, it is natural to consider the rational (or Hochschild) cohomology of G with coefficients in a k[G]comodule M. This is naturally isomorphic to the cohomology of the dual cocommutative Hopf algebra k[G] # with coefficients in the k[G] #module M. In this latter formulation, we encounter familiar examples of the cohomology of group algebras kπ of a finite groups π and of restricted enveloping algebras V (g) of finite dimensional restricted Lie algebras g. In recent years, the representation theory of the algebras kπ and V (g) has been studied by considering the spectrum of the cohomology algebra with coefficients in the ground field k and the support in this spectrum of the cohomology with coefficients in various modules. This approach relies on the fact that H ∗ (π, k) and H ∗ (V (g), k) are finitely generated kalgebras as proved in [G], [E], [V], [FP2]. Rational representations of algebraic groups in positive characteristic correspond to representations of a hierarchy of finite group schemes. In order to begin the process of introducing geometric methods to the study of these other group schemes, finite generation must be proved. Such a proof has proved surprisingly elusive (though partial results can be found in [FP2]). The main theorem of this paper is the following: Theorem 1.1. Let G be a finite group scheme and M a finite dimensional rational Gmodule. Then H ∗ (G, k) is a finitely generated kalgebra and H ∗ (G, M) is a finite H ∗ (G, k)module. Work in progress by C. Bendel (and the authors) reveals that Theorem 1.1 and its proof will provide interesting theorems of a geometric nature concerning the representation theory of finite group schemes. In a sense that is made explicit in section 1, our proof of finite generation is quite constructive. We embed G in some general linear group GLn and establish the existence of universal extension classes for GLn of specified degrees. In a direct manner, these classes provide the generators of H ∗ (G, k). In order to construct these universal extension classes, we follow closely the approach of V. Franjou, J. Lannes, and L. Schwartz [FLS]. This entails the study of
Stable Homotopy of Algebraic Theories
 Topology
, 2001
"... The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several familiar algebraic t ..."
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Cited by 11 (1 self)
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The simplicial objects in an algebraic category admit an abstract homotopy theory via a Quillen model category structure. We show that the associated stable homotopy theory is completely determined by a ring spectrum functorially associated with the algebraic theory. For several familiar algebraic theories we can identify the parameterizing ring spectrum; for other theories we obtain new examples of ring spectra. For the theory of commutative algebras we obtain a ring spectrum which is related to AndreH}Quillen homology via certain spectral sequences. We show that the (co)homology of an algebraic theory is isomorphic to the topological Hochschild (co)homology of the parameterizing ring spectrum. # 2000 Elsevier Science Ltd. All rights reserved. MSC: 55U35; 18C10 Keywords: Algebraic theories; Ring spectra; AndreH}Quillen homology; #spaces The original motivation for this paper came from the attempt to generalize a rational result about the homotopy theory of commutative rings. For...
Hodge Decomposition For Higher Order Hochschild Homology
"... this paper is to show that the higher order Hochschild homology in charecteristic zero case have naturaral decomposition, which for d = 1 is isomorphic to the classical Hodge decomposition ([L2]). Our methods are new even for d = 1 and are based on homological properties of \Gammamodules. ..."
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Cited by 9 (0 self)
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this paper is to show that the higher order Hochschild homology in charecteristic zero case have naturaral decomposition, which for d = 1 is isomorphic to the classical Hodge decomposition ([L2]). Our methods are new even for d = 1 and are based on homological properties of \Gammamodules.
Cohomology of Monoids in Monoidal Categories
 In "Operads: Proceedings of renaissance conferences." Contemp. Math. 202, AMS
, 1997
"... this article we show that these structures are still susceptible to cohomological investigation, by developing the theory in the absence of the symmetry condition. Later we shall assume that the monoidal structure is left distributive over coproducts and the category is an abelian category; this is ..."
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Cited by 6 (2 self)
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this article we show that these structures are still susceptible to cohomological investigation, by developing the theory in the absence of the symmetry condition. Later we shall assume that the monoidal structure is left distributive over coproducts and the category is an abelian category; this is the case for operads, our original motivating example. 1. Monoids and Modules We define monoids in monoidal categories and introduce the "module" objects which will be used later as coefficients in the cohomology of such monoids. We also give some of our motivating examples of monoidal categories and the monoids therein. Let us start by recalling that a monoidal category is a tuple V = (V; ffi; I; a; l; r) where V is a category, ffi : V \Theta V ! V is a functor, I is an object of V, and a = (a X;Y;Z : (X ffi Y ) ffi Z ! X ffi (Y ffi Z)) X;Y;Z2V ; l = (l X : I ffi X ! X)X2V ; r = (r X : X ffi I ! X)X2V are natural isomorphisms, required to satisfy certain conditions which we omit here (see e.g. [19]). In many examples our monoidal categories will be strictly associative and have strict units, in the sense that all aX;Y;Z and l X , r X are identity morphisms. The monoidal category V is abelian if the underlying category V is an abelian category. Suppose V has binary coproducts, denoted X t Y ; then the monoidal structure is left distributive if the canonical natural transformation (X 1 ffi Y ) t (X 2 ffi Y ) ! (X 1 t X 2 ) ffi Y is an isomorphism. Right distributivity is defined similarly. A strict monoidal functor between monoidal categories is a functor between the underlying categories preserving all the existing structure in the obvious way. Given such a V, a monoid in V, or a Vmonoid, is a triple G = (G; ; j) where G 2 V, : G ffi G ! G, j : I ! G must satisfy the...
Comparison of Abelian Categories Recollements
 DOCUMENTA MATH.
, 2004
"... We give a necessary and sufficient condition for a morphism between recollements of abelian categories to be an equivalence. ..."
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Cited by 4 (2 self)
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We give a necessary and sufficient condition for a morphism between recollements of abelian categories to be an equivalence.
Universal Toda brackets of ring spectra
, 2006
"... Abstract. We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of Rmodule spectra. It determines ..."
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Cited by 4 (1 self)
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Abstract. We construct and examine the universal Toda bracket of a highly structured ring spectrum R. This invariant of R is a cohomology class in the Mac Lane cohomology of the graded ring of homotopy groups of R which carries information about R and the category of Rmodule spectra. It determines for example all triple Toda brackets of R and the first obstruction to realizing a module over the homotopy groups of R by an Rmodule spectrum. For periodic ring spectra, we study the corresponding theory of higher universal Toda brackets. The real and complex Ktheory spectra serve as our main examples. 1.
Central Extensions and Nilpotence of Maltsev Theories
"... Relationship is clarified between the notions of linear extension of algebraic theories, and central extension, in the sense of commutator calculus, of their models. Varieties of algebras turn out to be nilpotent Maltsev precisely when their theories may be obtained as results of iterated linear ext ..."
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Cited by 3 (3 self)
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Relationship is clarified between the notions of linear extension of algebraic theories, and central extension, in the sense of commutator calculus, of their models. Varieties of algebras turn out to be nilpotent Maltsev precisely when their theories may be obtained as results of iterated linear extensions by bifunctors from the so called abelian theories. The latter theories are described; they are slightly more general than theories of modules over a ring.
Homotopy ends and Thomason model categories
 Selecta Math
"... In the last year of his life, Robert W. Thomason reworked the notion of a model category, used to adapt homotopy theory to algebra, and used homotopy ends to affirmatively solve a problem raised by Grothendieck: find a notion of model structure which is inherited by functor categories. The axioms fo ..."
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Cited by 3 (0 self)
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In the last year of his life, Robert W. Thomason reworked the notion of a model category, used to adapt homotopy theory to algebra, and used homotopy ends to affirmatively solve a problem raised by Grothendieck: find a notion of model structure which is inherited by functor categories. The axioms for such a Thomason model category were published later in [WT]. In this paper we explain and prove Thomason’s results, based on his private notebooks [T78]–[T86]. Each of Thomason’s 180page notebooks are spiralbound with an index on the inside cover. We have chosen to follow their internal citation method: our citation [Tx, y] refers to page y of Thomason’s notebook x. We first present Thomason’s ideas about homotopy ends and its generalizations (sections 1 and 2). These are first formulated for complete simplicial closed model categories in the sense of Quillen [QH], because of their usefulness in this setting, as demonstrated by Dwyer and Kan [DK]. Thomason’s axioms and examples come next, followed by a proof that the
SHUKLA COHOMOLOGY AND ADDITIVE TRACK THEORIES
, 2004
"... It is well known that the Hochschild cohomology for associative algebras has good properties only for algebras which are projective modules over the ground ring. For general algebras behavior of Hochschild cohomology is more pathological, for example there is no long cohomological exact sequence cor ..."
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Cited by 3 (0 self)
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It is well known that the Hochschild cohomology for associative algebras has good properties only for algebras which are projective modules over the ground ring. For general algebras behavior of Hochschild cohomology is more pathological, for example there is no long cohomological exact sequence corresponding to