Cohomology for Drinfeld doubles of some infinitesimal group schemes

Friedlander, Eric M.; Negron, Cris

doi:10.2140/ant.2018.12.1281

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Cohomology for Drinfeld doubles of some infinitesimal group schemes

Eric M. Friedlander and Cris Negron

Vol. 12 (2018), No. 5, 1281–1309

DOI: 10.2140/ant.2018.12.1281

Abstract

Consider a field $k$ of characteristic $p > 0$ , the $r$ -th Frobenius kernel $G_{(r)}$ of a smooth algebraic group $G$ , the Drinfeld double $D G_{(r)}$ of $G_{(r)}$ , and a finite dimensional $D G_{(r)}$ -module $M$ . We prove that the cohomology algebra $H^{*} (D G_{(r)}, k)$ is finitely generated and that $H^{*} (D G_{(r)}, M)$ is a finitely generated module over this cohomology algebra. We exhibit a finite map of algebras $θ_{r} : H^{*} (G_{(r)}, k) \otimes S (g) \to H^{*} (D G_{(r)}, k)$ , which offers an approach to support varieties for $D G_{(r)}$ -modules. For many examples of interest, $θ_{r}$ is injective and induces an isomorphism of associated reduced schemes. For $M$ an irreducible $D G_{(r)}$ -module, $θ_{r}$ enables us to identify the support variety of $M$ in terms of the support variety of $M$ viewed as a $G_{(r)}$ -module.

Keywords

Hopf cohomology, Drinfeld doubles, finite group schemes

Mathematical Subject Classification 2010

Primary: 57T05

Secondary: 20G10, 20G40

Milestones

Received: 9 October 2017

Revised: 12 February 2018

Accepted: 29 March 2018

Published: 31 July 2018

Authors

Eric M. Friedlander
Department of Mathematics University of Southern California Los Angeles, CA United States

Cris Negron
Department of Mathematics Massachusetts Institute of Technology Cambridge, MA United States

Vol. 12, No. 5, 2018