The cohomology algebra of a commutative group scheme

Let k be a commutative ring with unit of characteristic p > 0 and let G = Spec(A) be an affine commutative group scheme over k. Let H•(G) be the graded Hochschild algebraic group cohomology algebra and, for M a rational G-module, let H·(G, M) denote the graded Hochschild cohomology H•(G)-module. We show that H•(G) is, in general, a graded Hopf algebra. When G = G_{a, k}, let α_p^νdenote the subgroup of p^νnilpotents and let F_ν, denote the νth power of the Frobenius. We show that for any finite M that there is a ν such that H•(G_{a, k}, M)∜ H•(α_p^ν, M)⊗_kF*_ν(H•(G_{a, k})) where F*_ν is the endomorphism of H•(G_{a, k}) induced by F_ν. As a consequence, we can show that H• (G_{a, k}, M) is a finitely generated module over H•(G_{a, k}) when M is a finite dimensional vector space over k.