We give the first examples of finite groups such that the Chow ring of the classifying space 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 is mixed Tate, which is equivalent to the motivic Künneth property. We prove that is mixed Tate for various “well-behaved” finite groups , such as the finite general linear groups in cross-characteristic and the symmetric groups.
"The motive of a classifying space." Geom. Topol. 20 (4) 2079 - 2133, 2016. https://doi.org/10.2140/gt.2016.20.2079