Affine representability results in ${\mathbb A}^1$-homotopy theory I: vector bundles

TL;DR

This paper proves a broad affine representability theorem in ${ m A}^1$-homotopy theory and applies it to generalize vector bundle representability results, simplifying previous theorems and broadening their scope.

Contribution

It introduces a general affine representability result in ${ m A}^1$-homotopy theory and extends vector bundle representability, surpassing prior specific cases.

Findings

Established a general affine representability theorem.

Generalized and simplified ${ m A}^1$-representability for vector bundles.

Broadened applicability of vector bundle classification in algebraic geometry.

Abstract

We establish a general "affine representability" result in ${\mathbb A}^1$-homotopy theory over a general base. We apply this result to obtain representability results for vector bundles in ${\mathbb A}^1$-homotopy theory. Our results simplify and significantly generalize F. Morel's ${\mathbb A}^1$-representability theorem for vector bundles.