Support Theory for Extended Drinfeld Doubles

Abstract

Following earlier work with Cris Negron on the cohomology of Drinfeld doubles $D({\mathbb{G}}_{(r)})$, we develop a "geometric theory" of support varieties for "extended Drinfeld doubles" $\tilde{D}({\mathbb{G}}_{(r)})$ of Frobenius kernels ${\mathbb{G}}_{(r)}$ of smooth linear algebraic groups $\mathbb{G}$ over a field $k$ of characteristic $p>0$. To a $\tilde{D}({\mathbb{G}}_{(r)})$-module $M$ we associate the space $\Pi (\tilde{D}({\mathbb{G}}_{(r)}){)}_M$ of equivalence classes of "pairs of $\pi$-points" and prove most of the desired properties of $M\mapsto \Pi (\tilde{D}({\mathbb{G}}_{(r)}){)}_M$. Namely, this association satisfies the "tensor product property" and admits a natural continuous map ${\Psi }_{\tilde{D}}$ to cohomological support theory. Moreover, for $M$ finite dimensional and with suitable conditions on ${\mathbb{G}}_{(r)}$, this association provides a "projectivity test", ${\Psi }_{\tilde{D}}$ is a homeomorphism, and identifies $\Pi (\tilde{D}({\mathbb{G}}_{(r)}){)}_M$ with the cohomological support variety of $M$ for various classes of $\tilde{D}({\mathbb{G}}_{(r)})$-modules $M$.

Cited by in corpus (1)

• Support theory for Drinfeld doubles of some infinitesimal group schemes