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

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.