Quotient stacks and equivariant \'etale cohomology algebras: Quillen's theory revisited

TL;DR

This paper proves that the equivariant étale cohomology algebra of quotient stacks by algebraic group actions over an algebraically closed field is finitely generated, and establishes a Quillen-like structure theorem involving elementary abelian -subgroups.

Contribution

It introduces a finiteness result for equivariant étale cohomology algebras and a structure theorem generalizing Quillen's theorem to quotient stacks.

Findings

Equivariant étale cohomology algebras are finitely generated over the coefficient ring.

A structure theorem relates cohomology to fixed points of elementary abelian -subgroups.

Analysis of specialization of points is key to the proof.

Abstract

Let $k$ be an algebraically closed field. Let $\Lambda$ be a noetherian commutative ring annihilated by an integer invertible in $k$ and let $\ell$ be a prime number different from the characteristic of $k$. We prove that if $X$ is a separated algebraic space of finite type over $k$ endowed with an action of a $k$-algebraic group $G$, the equivariant \'etale cohomology algebra $H^*([X/G],\Lambda)$, where $[X/G]$ is the quotient stack of $X$ by $G$, is finitely generated over $\Lambda$. Moreover, for coefficients $K \in D^+_c([X/G],\mathbb{F}_{\ell})$ endowed with a commutative multiplicative structure, we establish a structure theorem for $H^*([X/G],K)$, involving fixed points of elementary abelian $\ell$-subgroups of $G$, which is similar to Quillen's theorem in the case $K = \mathbb{F}_{\ell}$. One key ingredient in our proof of the structure theorem is an analysis of specialization of points of the quotient stack. We also discuss variants and generalizations for certain Artin stacks.