Affine representability results in A^1-homotopy theory II: principal bundles and homogeneous spaces

TL;DR

This paper extends affine representability results in ${f A}^1$-homotopy theory to principal bundles and homogeneous spaces, establishing invariance and representability theorems for torsors under reductive groups.

Contribution

It introduces a relative affine representability theorem and proves ${f A}^1$-invariance for torsors, advancing the understanding of principal bundles in algebraic geometry.

Findings

Proves a relative version of affine representability in ${f A}^1$-homotopy theory.

Establishes ${f A}^1$-invariance for generically trivial torsors under isotropic reductive groups.

Derives representability theorems for torsors and homogeneous spaces in ${f A}^1$-homotopy theory.

Abstract

We establish a relative version of the abstract "affine representability" theorem in ${\mathbb A}^1$--homotopy theory from Part I of this paper. We then prove some ${\mathbb A}^1$--invariance statements for generically trivial torsors under isotropic reductive groups over infinite fields analogous to the Bass--Quillen conjecture for vector bundles. Putting these ingredients together, we deduce representability theorems for generically trivial torsors under isotropic reductive groups and for associated homogeneous spaces in ${\mathbb A}^1$--homotopy theory.