A Friedlander-Suslin theorem over a noetherian base ring

TL;DR

This paper proves that the cohomology algebra of a finite flat group scheme acting on a finitely generated algebra over a noetherian base ring is finitely generated, unifying previous results in the area.

Contribution

It extends the Friedlander-Suslin theorem to a broader setting over noetherian base rings, establishing finite generation of cohomology algebras.

Findings

Cohomology algebra $H^*(G,A)$ is finitely generated over $k$

Unifies earlier results on cohomology finiteness

Applicable to finite flat group schemes over noetherian rings

Abstract

Let $k$ be a noetherian commutative ring and let $G$ be a finite flat group scheme over $k$. Let $G$ act rationally on a finitely generated commutative $k$-algebra $A$. We show that the cohomology algebra $H^*(G,A)$ is a finitely generated $k$-algebra. This unifies some earlier results.