Finite generation of cohomology for Drinfeld doubles of finite group schemes

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 \({{\,\mathrm{rep}\,}}(G)^*_\mathscr {M}\) dual to \({{\,\mathrm{rep}\,}}(G)\) are 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 Friedlander and the author.