The Cohomology Ring of a Finite Group Scheme

Abstract

Let $k$ be a field and let $A$ be a $k$-algebra with additional structure so that Spec $A$ is a finite commutative group scheme over $k$, (so $A$ is a Hopf algebra). Let $H^\bullet (A,k)$ be the Hochschild cohomology ring. In another paper, we demonstrated that if $k$ is a perfect field: (a) $H^\bullet (A,k)$ is generated by ${H^1}$ and $H_{\operatorname {sym} }^2$. (b) If characteristic $k = p \ne 2$, then $H^\bullet (A,k)$ is freely generated by ${H^1}$ and $H_{\operatorname {sym} }^2$. (c) If characteristic $k = 2$, then there are subspaces ${V_1},{V_2}$ of ${H^1}$ and ${V_3}$ of $H_{\operatorname {sym} }^2$ such that $H^\bullet (A,k)$ is generated by ${V_1},{V_2},{V_3}$ and the only relations are ${f^2} = 0$ for all $f$ in ${V_1}$. In this paper we show that if $k$ is arbitrary (a) and (b) still hold, and we use an example of Oort and Mumford to show that (c) does not hold for arbitrary $k$.