Vol. 12, No. 5, 2018

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Cohomology for Drinfeld doubles of some infinitesimal group schemes

Eric M. Friedlander and Cris Negron

Vol. 12 (2018), No. 5, 1281–1309

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.

Hopf cohomology, Drinfeld doubles, finite group schemes
Mathematical Subject Classification 2010
Primary: 57T05
Secondary: 20G10, 20G40
Received: 9 October 2017
Revised: 12 February 2018
Accepted: 29 March 2018
Published: 31 July 2018
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