Finite generation of cohomology for Drinfeld doubles of finite group schemes

TL;DR

This paper proves that the cohomology of Drinfeld doubles of finite group schemes is finitely generated, extending previous results and providing bounds on the structure of related categories.

Contribution

It establishes finite generation of cohomology for Drinfeld doubles of any finite group scheme, a significant advancement in understanding their algebraic properties.

Findings

Cohomology of D(G) is finitely generated.

Extensions between trivial and other representations are finitely generated modules.

Categories dual to rep(G) have finite type with bounded Krull dimension.

Abstract

We prove that the Drinfeld double of an arbitrary finite group scheme has finitely generated cohomology. That is to say, for G any finite group scheme, and D(G) the Drinfeld double of the group ring kG, we show that the self-extension algebra of the trivial representation for D(G) is a finitely generated algebra, and that for each D(G)-representation V the extensions from the trivial representation to V form a finitely generated module over the aforementioned algebra. As a corollary, we find that all categories rep(G)*_M dual to rep(G) are of also of finite type (i.e. have finitely generated cohomology), and we provide a uniform bound on their Krull dimensions. This paper completes earlier work of E. M. Friedlander and the author.