Cohomology for Drinfeld doubles of some infinitesimal group schemes

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

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