A Geometric Splitting of the Motive of $\textrm{GL}_n$

TL;DR

This paper establishes an algebraic splitting of the motive of the general linear group in Voevodsky's motivic category, paralleling a topological splitting involving Thom spaces over Grassmannians, thus bridging algebraic and topological motivic structures.

Contribution

It provides an algebraic analog of Miller's topological splitting of unitary groups, demonstrating a similar decomposition of the motive of GL_n in the motivic category.

Findings

Existence of algebraic Thom space analogs in motivic categories

Splitting of the motive M(GL_n) in DM(k,R)

Parallel between algebraic and topological motivic decompositions

Abstract

A paper by Haynes Miller shows that there is a filtration on the unitary groups that splits in the stable homotopy category, where the stable summands are certain Thom spaces over Grassmannians. We give an algebraic version of this result in the context of Voevodsky's tensor triangulated category of stable motivic complexes $\textbf{DM}(k,R)$, where $k$ is a field. Specifically, we show that there are algebraic analogs of the Thom spaces appearing in Miller's splitting that give rise to an analogous splitting of the motive $M(\textrm{GL}_n)$ in $\textbf{DM}(k,R)$, where $\textrm{GL}_n$ is the general linear group scheme over $k$.