#### Volume 20, issue 4 (2016)

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The motive of a classifying space

### Burt Totaro

Geometry & Topology 20 (2016) 2079–2133
##### Abstract

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 $ℚ$. 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 $BG$ 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.

##### Keywords
Chow ring, mixed Tate motive, classifying space
##### Mathematical Subject Classification 2010
Primary: 14C15
Secondary: 14F42, 14M20, 14A20