Bifunctor cohomology and cohomological finite generation for reductive groups

Abstract

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 $A^G=H^0(G,A)$ 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 ${\Gamma }^{*}({\mathfrak{gl}}^{(1)})$