The mixed Tate property of reductive groups

TL;DR

This thesis investigates the mixed Tate property of reductive algebraic groups, establishing conditions under which classifying spaces have Chow Kunneth properties and constructing motives for quotient stacks.

Contribution

It refines the construction of compactly supported motives for algebraic spaces and quotient stacks, and analyzes the mixed Tate property for classical and exceptional groups.

Findings

All split forms of classical and G2 groups satisfy the mixed Tate property.

Non-split forms of these groups do not satisfy the property.

Motives of classifying spaces are isomorphic under certain group extensions.

Abstract

This thesis is concerned with the mixed Tate property of reductive algebraic groups $G$, which in particular guarantees a Chow Kunneth property for the classifying space $BG$. Toward this goal, we first refine the construction of the compactly supported motive of a quotient stack. In the first section, we construct the compactly supported motive $M^c(X)$ of an algebraic space $X$ and demonstrate that it satisfies expected properties, following closely Voevodsky's work in the case of schemes. In the second section, we construct a functorial version of Totaro's definition of the compactly supported motive $M^c([X/G])$ for any quotient stack $[X/G]$ where $X$ is an algebraic space and $G$ is an affine group scheme acting on it. A consequence of functoriality is a localization triangle for these motives. In the third section, we study the mixed Tate property for the classical groups as well as the exceptional group $G_2$. For these groups, we demonstrate that all split forms satisfy the mixed Tate property, while exhibiting non-split forms that do not. Finally, we prove that for any affine group scheme $G$ and normal split unipotent subgroup $J$ of $G$, the motives $M^c(BG)$ and $M^c(B(G/J))$ are isomorphic.