Affine representability results in A^1-homotopy theory III: finite fields and complements

TL;DR

This paper extends affine representability results in ${ m A}^1$-homotopy theory to finite fields, providing streamlined proofs and analyzing fiber sequences and motivic spheres as homogeneous spaces.

Contribution

It offers a simplified proof of ${ m A}^1$-representability for $G$-torsors over finite fields and identifies new motivic spheres as homogeneous spaces.

Findings

Extended ${ m A}^1$-representability to finite fields

Analyzed fiber sequences in ${ m A}^1$-homotopy theory

Identified final examples of motivic spheres as homogeneous spaces

Abstract

We give a streamlined proof of ${\mathbb A}^1$-representability for $G$-torsors under "isotropic" reductive groups, extending previous results in this sequence of papers to finite fields. We then analyze a collection of group homomorphisms that yield fiber sequences in ${\mathbb A}^1$-homotopy theory, and identify the final examples of motivic spheres that arise as homogeneous spaces for reductive groups.