Cohomological finite generation for the group scheme $SL_2$

TL;DR

This paper proves that the cohomology ring of the group scheme SL_2 acting on finitely generated algebras over a noetherian ring is itself finitely generated, extending understanding of algebraic group actions.

Contribution

It establishes the finite generation of the cohomology ring for the group scheme SL_2 acting on finitely generated algebras over noetherian rings, a significant advancement in algebraic group cohomology.

Findings

H^*(G,A) is finitely generated as a k-algebra

Applicable to group scheme SL_2 over noetherian rings

Extends cohomological finite generation results

Abstract

Let $G$ be the group scheme $SL_2$ defined over a noetherian ring $k$. If $G$ acts on a finitely generated commutative $k$-algebra $A$, then $H^*(G,A)$ is a finitely generated $k$-algebra.