Cohomology for Drinfeld doubles of some infinitesimal group schemes

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.