Bifunctor cohomology and Cohomological finite generation for reductive groups

Let G be a reductive linear algebraic group over a field k. Let A be a finitely generated commutative k-algebra on which G acts rationally by k-algebra automorphisms. Invariant theory tells that the ring of invariants A^G=H^0(G,A) is finitely generated. We show that in fact the full cohomology ring H^*(G,A) is finitely generated. The proof is based on the strict polynomial bifunctor cohomology classes constructed by the junior author. We also continue the study of bifunctor cohomology of the divided powers of a Frobenius twist of the adjoint representation.