Cohomology for Drinfeld Doubles of Some Infinitesimal Group Schemes

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 θr : H(G(r), k) ⊗ S(g) → H(DG(r), k) which offers an approach to support varieties for DG(r)-modules. For many examples of interest, θr is injective and induces an isomorphism of associated reduced schemes. Additionally, for M an irreducible DG(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.