Bifunctor cohomology and Cohomological finite generation for reductive groups

International audienceLet G be a reductive linear algebraic group over afield 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 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 in [22]. We also continue the study of bifunctor cohomology of Gamma*(gl((1)))