Finite generation of cohomology for Drinfeld doubles of finite group schemes

Cris Negron ^[1]
1. [1] University of North Carolina, USA
Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 27, Nº. 2, 2021
Idioma: inglés
DOI: 10.1007/s00029-021-00637-2
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Resumen
- 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 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.