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Consider a field k of characteristic p > 0, G_r the r-th Frobenius kernel ofa smooth algebraic group G, DG_r the Drinfeld double of G_r, and M a finitedimensional DG_r-module. We prove that the cohomology algebra H*(DG_r,k) isfinitely generated and that H*(DG_r,M) is a finitely generated module over thiscohomology 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 forDG_r-modules. For many examples of interest, \theta_r is injective and inducesan isomorphism of associated reduced schemes. Additionally, for M anirreducible DG_r-module, \theta_r enables us to identify the support variety ofM in terms of the support variety of M viewed as a G_r-module.

Cohomology for Drinfeld doubles of some infinitesimal group schemes