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2016 The motive of a classifying space
Burt Totaro
Geom. Topol. 20(4): 2079-2133 (2016). DOI: 10.2140/gt.2016.20.2079


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.


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Burt Totaro. "The motive of a classifying space." Geom. Topol. 20 (4) 2079 - 2133, 2016.


Received: 27 November 2014; Accepted: 12 September 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1375.14027
MathSciNet: MR3548464
Digital Object Identifier: 10.2140/gt.2016.20.2079

Primary: 14C15
Secondary: 14A20 , 14F42 , 14M20

Keywords: Chow ring , classifying space , mixed Tate motive

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.20 • No. 4 • 2016
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