Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Quotient stacks and equivariant étale cohomology algebras: Quillen’s theory revisited

Authors: Luc Illusie and Weizhe Zheng
Journal: J. Algebraic Geom. 25 (2016), 289-400
Published electronically: February 10, 2016
MathSciNet review: 3466353
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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 $H^*([X/G],\Lambda )$, 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.

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Additional Information

Luc Illusie
Affiliation: Laboratoire de Mathématiques d’Orsay, Université Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France
MR Author ID: 90990
ORCID: 0000-0002-6634-6325

Weizhe Zheng
Affiliation: Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
MR Author ID: 733496

Received by editor(s): May 31, 2013
Received by editor(s) in revised form: November 12, 2013, July 29, 2014, November 5, 2014, and February 8, 2015
Published electronically: February 10, 2016
Additional Notes: The second author was partially supported by China’s Recruitment Program of Global Experts; National Natural Science Foundation of China Grant 11321101; Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences; National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences.
Dedicated: To the memory of Daniel Quillen
Article copyright: © Copyright 2016 University Press, Inc.