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 states that the ring of invariants AG=H0(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 in [22]. We also continue the study of bifunctor cohomology of Γ∗(gl(1)