Quotient stacks and equivariant étale 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 étale cohomology algebra $H^{\ast }([X/G],\Lambda )$, 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 $H^{\ast }([X/G],K)$, 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.

76 pages. v5: fixed typos

References in corpus (4)

• Revêtements étales et groupe fondamental (SGA 1)
• Odds and ends on finite group actions and traces
• Existence and properties of geometric quotients
• Enhanced six operations and base change theorem for higher Artin stacks

Cited by in corpus (2)

• Duality and nearby cycles over general bases
• Companions on Artin stacks