The cohomology ring of a finite group scheme

Abstract

Let k k be a field and let A A be a k k -algebra with additional structure so that Spec A A is a finite commutative group scheme over k k , (so A A is a Hopf algebra). Let H ∙ ( A , k ) H^\bullet (A,k) be the Hochschild cohomology ring. In another paper, we demonstrated that if k k is a perfect field: (a) H ∙ ( A , k ) H^\bullet (A,k) is generated by H 1 {H^1} and H sym 2 H_{\operatorname {sym} }^2 . (b) If characteristic k = p ≠ 2 k = p \ne 2 , then H ∙ ( A , k ) H^\bullet (A,k) is freely generated by H 1 {H^1} and H sym 2 H_{\operatorname {sym} }^2 . (c) If characteristic k = 2 k = 2 , then there are subspaces V 1 , V 2 {V_1},{V_2} of H 1 {H^1} and V 3 {V_3} of H sym 2 H_{\operatorname {sym} }^2 such that H ∙ ( A , k ) H^\bullet (A,k) is generated by V 1 , V 2 , V 3 {V_1},{V_2},{V_3} and the only relations are f 2 = 0 {f^2} = 0 for all f f in V 1 {V_1} . In this paper we show that if k 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 k .