The motive of a classifying space

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 $\mathbb{Q}$. 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.