The motive of a classifying space

TL;DR

This paper explores the dependence of the Chow ring of classifying spaces on the base field, providing new examples and characterizations, and investigates when these motives are mixed Tate, with implications for algebraic geometry and motivic theory.

Contribution

It introduces the first examples of finite groups with classifying space Chow rings dependent on the base field and characterizes varieties satisfying Kunneth properties for Chow groups.

Findings

Chow ring of BG depends on the base field for certain finite groups.

BG is mixed Tate for groups like finite general linear groups and symmetric groups.

Provides characterizations of varieties with Kunneth properties for Chow groups.

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 which satisfy Kunneth 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 Kunneth 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.