Cohomology for Drinfeld doubles of some infinitesimal group schemes

TL;DR

This paper proves finite generation of cohomology algebras for Drinfeld doubles of Frobenius kernels of algebraic groups over fields of positive characteristic, and relates support varieties of modules to those of the underlying group schemes.

Contribution

It establishes finite generation of cohomology for DG_r and introduces a map connecting cohomology of G_r and DG_r, aiding in support variety analysis.

Findings

Cohomology algebra H*(DG_r,k) is finitely generated.

H*(DG_r,M) is finitely generated over H*(DG_r,k).

Support varieties of modules are characterized via a finite algebra map.

Abstract

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