The motive of a classifying space

Totaro, Burt

doi:10.2140/gt.2016.20.2079

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The motive of a classifying space

Burt Totaro

Geometry & Topology 20 (2016) 2079–2133

DOI: 10.2140/gt.2016.20.2079

Abstract

We give the first examples of finite groups $G$ such that the Chow ring of the classifying space $B G$ depends on the base field, even for fields containing the algebraic closure of $ℚ$ . As a tool, we give several characterizations of the varieties that satisfy Künneth properties for Chow groups or motivic homology.

We define the (compactly supported) motive of a quotient stack in Voevodsky’s derived category of motives. This makes it possible to ask when the motive of $B G$ is mixed Tate, which is equivalent to the motivic Künneth property. We prove that $B G$ is mixed Tate for various “well-behaved” finite groups $G$ , such as the finite general linear groups in cross-characteristic and the symmetric groups.

Keywords

Chow ring, mixed Tate motive, classifying space

Mathematical Subject Classification 2010

Primary: 14C15

Secondary: 14F42, 14M20, 14A20

References

Publication

Received: 27 November 2014

Accepted: 12 September 2015

Published: 15 September 2016

Proposed: Lothar Göttsche

Seconded: Jim Bryan, Frances Kirwan

Authors

Burt Totaro
Department of Mathematics University of California, Los Angeles Box 951555 Los Angeles, CA 90095-1555 United States
http://www.math.ucla.edu/~totaro/

Volume 20, issue 4 (2016)