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 \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.