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The motive of a classifying space

Burt Totaro

Geometry & Topology 20 (2016) 2079–2133

We give the first examples of finite groups G such that the Chow ring of the classifying space BG depends on the base field, even for fields containing the algebraic closure of . 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 BG is mixed Tate, which is equivalent to the motivic Künneth property. We prove that BG is mixed Tate for various “well-behaved” finite groups G, such as the finite general linear groups in cross-characteristic and the symmetric groups.

Chow ring, mixed Tate motive, classifying space
Mathematical Subject Classification 2010
Primary: 14C15
Secondary: 14F42, 14M20, 14A20
Received: 27 November 2014
Accepted: 12 September 2015
Published: 15 September 2016
Proposed: Lothar Göttsche
Seconded: Jim Bryan, Frances Kirwan
Burt Totaro
Department of Mathematics
University of California, Los Angeles
Box 951555
Los Angeles, CA 90095-1555
United States