Support Theory for Extended Drinfeld Doubles

TL;DR

This paper develops a geometric support theory for extended Drinfeld doubles of Frobenius kernels, linking module properties with cohomological support varieties through a new framework involving pairs of pi-points.

Contribution

It introduces a support variety theory for extended Drinfeld doubles, establishing key properties and connections to cohomology, including a projectivity test and a homeomorphism with support varieties.

Findings

Support varieties satisfy the tensor product property.

The support map is a homeomorphism under certain conditions.

Provides a new tool for analyzing modules over 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 $M$.