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Register a new userFinite generation of cohomology for Drinfeld doubles of finite group schemes
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Authors
Cris Negron
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We prove that the Drinfeld double of an arbitrary finite group scheme has finitely generated cohomology. That is to say, for G any finite group scheme, and D(G) the Drinfeld double of the group ring kG, we show that the self-extension algebra of the trivial representation for D(G) is a finitely generated algebra, and that for each D(G)-representation V the extensions from the trivial representation to V form a finitely generated module over the aforementioned algebra. As a corollary, we find that all categories rep(G)*_M dual to rep(G) are of also of finite type (i.e. have finitely generated cohomology), and we provide a uniform bound on their Krull dimensions. This paper completes earlier work of E. M. Friedlander and the author.
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Consider a Frobenius kernel G in a split semisimple algebraic group, in very good characteristic. We provide an analysis of support for the Drinfeld center Z(rep(G)) of the representation category for G, or equivalently for the representation category of the Drinfeld double of kG. We show that thick ideals in the corresponding stable category are classified by cohomological support, and calculate the Balmer spectrum of the stable category of Z(rep(G)). We also construct a $pi$-point style rank variety for the Drinfeld double, identify $pi$-point support with cohomological support, and show that both support theories satisfy the tensor product property. Our results hold, more generally, for Drinfeld doubles of Frobenius kernels in any smooth algebraic group which admits a quasi-logarithm, such as a Borel subgroup in a split semisimple group in very good characteristic.
We consider the finite generation property for cohomology of a finite tensor category C, which requires that the self-extension algebra of the unit Ext*_C(1,1) is a finitely generated algebra and that, for each object V in C, the graded extension group Ext*_C(1,V) is a finitely generated module over the aforementioned algebra. We prove that this cohomological finiteness property is preserved under duality (with respect to exact module categories) and taking the Drinfeld center, under suitable restrictions on C. For example, the stated result holds when C is a braided tensor category of odd Frobenius-Perron dimension. By applying our general results, we obtain a number of new examples of finite tensor categories with finitely generated cohomology. In characteristic 0, we show that dynamical quantum groups at roots of unity have finitely generated cohomology. We also provide a new class of examples in finite characteristic which are constructed via infinitesimal group schemes.
We prove various finiteness and representability results for flat cohomology of finite flat abelian group schemes. In particular, we show that if $f:Xrightarrow mathrm{Spec} (k)$ is a projective scheme over a field $k$ and $G$ is a finite flat abelian group scheme over $X$ then $R^if_*G$ is an algebraic space for all $i$. More generally, we study the derived pushforwards $R^if_*G$ for $f:Xrightarrow S$ a projective morphism and $G$ a finite flat abelian group scheme over $X$. We also develop a theory of compactly supported cohomology for finite flat abelian group schemes, describe cohomology in terms of the cotangent complex for group schemes of height $1$, and relate the Dieudonne modules of the group schemes $R^if_*mu _p$ to cohomology generalizing work of Illusie. A higher categorical version of our main representability results is also presented.
The Shapovalov determinant for a class of pointed Hopf algebras is calculated, including quantized enveloping algebras, Lusztigs small quantum groups, and quantized Lie superalgebras. Our main tools are root systems, Weyl groupoids, and Lusztig type isomorphisms. We elaborate powerful novel techniques for the algebras at roots of unity, and pass to the general case using a density argument. Key words: Hopf algebra, Nichols algebra, quantum group, representation
This paper generalizes Huangs cohomology theory of grading-restricted vertex algebras to meromorphic open-string vertex algebras (MOSVAs hereafter), which are noncommutative generalizations of grading-restricted vertex algebras introduced by Huang. The vertex operators for these algebras satisfy associativity but do not necessarily satisfy the commutativity. Moreover, the MOSVA and its bimodules considered in this paper do not necessarily have finite-dimensional homogeneous subspaces, though we do require that they have lower-bounded gradings. The construction and results in this paper will be used in a joint paper by Huang and the author to give a cohomological criterion of the reductivity for modules for grading-restricted vertex algebras
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