On the mixed Tate property and the motivic class of the classifying stack of a finite group

Scavia, Federico

doi:10.2140/ant.2022.16.2265

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On the mixed Tate property and the motivic class of the classifying stack of a finite group

Federico Scavia

Vol. 16 (2022), No. 10, 2265–2287

DOI: 10.2140/ant.2022.16.2265

Abstract

Let $G$ be a finite group, and let ${B_{ℂ} G}$ the class of its classifying stack $B_{ℂ} G$ in Ekedahl’s Grothendieck ring of algebraic $ℂ$ -stacks $K_{0} ({Stacks}_{ℂ})$ . We show that if $B_{ℂ} G$ has the mixed Tate property, the invariants $H^{i} ({B_{ℂ} G})$ defined by Ekedahl are zero for all $i \neq 0$ . We also extend Ekedahl’s construction of these invariants to fields of positive characteristic.

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Keywords

mixed Tate, Grothendieck ring, classifying stack, algebraic group

Mathematical Subject Classification

Primary: 14A20, 14C15

Secondary: 14J10

Milestones

Received: 8 August 2020

Revised: 2 December 2021

Accepted: 1 February 2022

Published: 28 January 2023

Authors

Federico Scavia
Department of Mathematics University of California Los Angeles, CA United States

Vol. 16, No. 10, 2022