Bifunctor cohomology and Cohomological finite generation for reductive groups

TL;DR

This paper proves that the full cohomology ring of a reductive algebraic group acting on a finitely generated algebra is finitely generated, extending classical invariant theory results using bifunctor cohomology methods.

Contribution

It establishes the finite generation of the entire cohomology ring for reductive groups acting on finitely generated algebras, utilizing bifunctor cohomology classes.

Findings

The cohomology ring H^*(G,A) is finitely generated.

Construction of strict polynomial bifunctor cohomology classes.

Extended study of bifunctor cohomology of Frobenius twists.

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 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.