Cohomology of finite group schemes over a field

A finite group scheme G over a field k is equivalent to its coordinate algebra, a finite dimensional commutative Hopf algebra k[G] over k. In many contexts, it is natural to consider the rational (or Hochschild) cohomology of G with coefficients in a k[G]-comodule M . This is naturally isomorphic to the cohomology of the dual cocommutative Hopf algebra k[G] with coefficients in the k[G]-module M . In this latter formulation, we encounter familiar examples of the cohomology of group algebras kπ of a finite groups π and of restricted enveloping algebras V (g) of finite dimensional restricted Lie algebras g. In recent years, the representation theory of the algebras kπ and V (g) has been studied by considering the spectrum of the cohomology algebra with coefficients in the ground field k and the support in this spectrum of the cohomology with coefficients in various modules. This approach relies on the fact that H(π, k) and H(V (g), k) are finitely generated k-algebras as proved in [G], [E], [V], [FP2]. Rational representations of algebraic groups in positive characteristic correspond to representations of a hierarchy of finite group schemes. In order to begin the process of introducing geometric methods to the study of these other group schemes, finite generation must be proved. Such a proof has proved surprisingly elusive (though partial results can be found in [FP2]). The main theorem of this paper is the following: