Vol. 12, No. 5, 2018

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Cohomology for Drinfeld doubles of some infinitesimal group schemes

Eric M. Friedlander and Cris Negron

Vol. 12 (2018), No. 5, 1281–1309
Abstract

Consider a field k of characteristic p > 0, the r-th Frobenius kernel G(r) of a smooth algebraic group G, the Drinfeld double D G(r) of G(r), and a finite dimensional D G(r)-module M. We prove that the cohomology algebra H(D G(r),k) is finitely generated and that H(D G(r),M) is a finitely generated module over this cohomology algebra. We exhibit a finite map of algebras θr : H(G(r),k) S(g) H(D G(r),k), which offers an approach to support varieties for D G(r)-modules. For many examples of interest, θr is injective and induces an isomorphism of associated reduced schemes. For M an irreducible D G(r)-module, θr enables us to identify the support variety of M in terms of the support variety of M viewed as a G(r)-module.

Keywords
Hopf cohomology, Drinfeld doubles, finite group schemes
Mathematical Subject Classification 2010
Primary: 57T05
Secondary: 20G10, 20G40
Milestones
Received: 9 October 2017
Revised: 12 February 2018
Accepted: 29 March 2018
Published: 31 July 2018
Authors
Eric M. Friedlander
Department of Mathematics
University of Southern California
Los Angeles, CA
United States
Cris Negron
Department of Mathematics
Massachusetts Institute of Technology
Cambridge, MA
United States