Quotient Stacks And Equivariant étale Cohomology Algebras ... - ArXiv

Mathematics > Algebraic Geometry arXiv:1305.0365 (math) [Submitted on 2 May 2013 (v1), last revised 10 Feb 2016 (this version, v5)] Title:Quotient stacks and equivariant étale cohomology algebras: Quillen's theory revisited Authors:Luc Illusie, Weizhe Zheng View a PDF of the paper titled Quotient stacks and equivariant \'etale cohomology algebras: Quillen's theory revisited, by Luc Illusie and Weizhe Zheng View PDF

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.

Comments:	76 pages. v5: fixed typos
Subjects:	Algebraic Geometry (math.AG)
MSC classes:	14F20 (Primary), 14F43, 14L15, 14L30, 20G10, 20J06, 55R40, 55S05 (Secondary)
Cite as:	arXiv:1305.0365 [math.AG]
(or arXiv:1305.0365v5 [math.AG] for this version)
https://doi.org/10.48550/arXiv.1305.0365 Focus to learn more arXiv-issued DOI via DataCite
Journal reference:	J. Algebraic Geom. (2016), no. 2, pp. 289-400
Related DOI:	https://doi.org/10.1090/jag/674 Focus to learn more DOI(s) linking to related resources

Submission history

From: Weizhe Zheng [view email] [v1] Thu, 2 May 2013 08:16:38 UTC (83 KB) [v2] Thu, 30 May 2013 13:08:09 UTC (84 KB) [v3] Tue, 29 Jul 2014 11:05:00 UTC (86 KB) [v4] Fri, 6 Feb 2015 11:43:26 UTC (90 KB) [v5] Wed, 10 Feb 2016 12:37:23 UTC (90 KB) Full-text links:

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