Certain cohomology rings of finite and formal group schemes

Abstract

Introduction and summary. In [2, I], Demazure defines the cohomology of groups in arbitrary categories. He starts with a V and immediately goes over to the category @' of contravariant functors V Sets. For a group G in l and a G-module F in @ (for the definition of these concepts see ?1) Demazure can define H-(G, F), the graded cohomology module. In ?1 we repeat this definition and also give the obvious generalization to the case where F is a ring in @, then H'(G, F) becomes a graded ring. Then in ?2, we consider the case of W, the of finite affine schemes over a commutative ring k. When (9 is the ring in le defined by (9(Spec R) = R, G is a group scheme, and G acts trivially on (9; then H'(G, (9) is canonically isomorphic to the Hochshild cohomology H-(A, k) for A the affine ring of the Cartier dual of G. In fact Theorem 2.1 even connects complexes used to define H'(G, (9) and H'(A, k). In Proposition 2.4 a familiar computation is carried out which enables us to compute H(A, k) in certain cases. The theorem of Dieudonne-Cartier [2, VIIb] shows that the above mentioned special case is sufficient to compute H'(G, (9) when k is a perfect field. Finally in ?3, the case of formal Lie groups is considered. If G is a divisible formal Lie group over a perfect field k of characteristic p>2 with connected dual, (9 the ring element: (9(Spf A)=A, and G acts trivially on (9, then