#### Vol. 12, No. 5, 2018

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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
##### Abstract

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