A Friedlander-Suslin Theorem over a Noetherian Base Ring

Let k be a noetherian commutative ring and let G be a finite flat group scheme over k. Let G act rationally on a finitely generated commutative k-algebra A. We show that the cohomology algebra H^∗(G, A) is a finitely generated k-algebra. This unifies some earlier results: If G is a constant group scheme, then it is a theorem of Evens (Trans. Amer. Math. Soc. 101, 224–239, 1961, Theorem 8.1), and if k is a field of finite characteristic, then it is a theorem of Friedlander and Suslin (Invent. Math. 127, 209–270, 1997). If k is a field of characteristic zero, then there is no higher cohomology, so then it is a theorem of invariant theory.