The cohomology algebra of a commutative group scheme

Fossum, Robert; Haboush, William

doi:10.1090/S0002-9947-1993-1112374-8

The cohomology algebra of a commutative group scheme
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by Robert Fossum and William Haboush PDF

Trans. Amer. Math. Soc. 339 (1993), 553-565 Request permission

Abstract:

Let $k$ be a commutative ring with unit of characteristic $p > 0$ and let $G = \operatorname {Spec}(A)$ be an affine commutative group scheme over $k$. Let ${{\text {H}}^ \bullet }(G)$ be the graded Hochschild algebraic group cohomology algebra and, for $M$ a rational $G$-module, let ${{\text {H}}^ \bullet }(G,M)$ denote the graded Hochschild cohomology ${{\text {H}}^ \bullet }(G)$-module. We show that ${{\text {H}}^ \bullet }(G)$ is, in general, a graded Hopf algebra. When $G = {{\mathbf {G}}_{a,k}}$, let ${\alpha _{{p^\nu }}}$ denote the subgroup of ${p^\nu }$-nilpotents and let ${{\text {F}}_\nu }$ denote the $\nu$th power of the Frobenius. We show that for any finite $M$ that there is a $\nu$ such that \[ {{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}},M) \cong {{\text {H}}^ \bullet }({\alpha _{{p^\nu }}},M){ \otimes _k}{\text {F}}_{\nu }^\ast ({{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}}))\] where ${\text {F}}_\nu ^\ast$ is the endomorphism of ${{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}})$ induced by ${F_v}$. As a consequence, we can show that ${{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}},M)$ is a finitely generated module over ${{\text {H}}^ \bullet }({{\mathbf {G}}_{a,k}})$ when $M$ is a finite dimensional vector space over $k$.

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