Homotopy Classification of Line Bundles Over Rigid Analytic Varieties

TL;DR

This paper develops a motivic homotopy theory for rigid analytic varieties and uses it to classify line bundles and vector bundles over quasi-Stein spaces, linking homotopy invariance to a rigid analytic version of the Bass--Quillen conjecture.

Contribution

It constructs a new motivic homotopy framework for rigid analytic varieties and establishes homotopy classifications of line and vector bundles within this setting.

Findings

Homotopy classification of line bundles by infinite projective space.

A homotopy invariance property equivalent to a rigid analytic Bass--Quillen conjecture.

Development of a motivic homotopy theory for rigid analytic varieties.

Abstract

We construct a motivic homotopy theory for rigid analytic varieties with the rigid analytic affine line $\mathbb{A} ^1_\mathrm{rig}$ as an interval object. This motivic homotopy theory is inspired from, but not equal to, Ayoub's motivic homotopy theory for rigid analytic varieties. Working in the so constructed homotopy theory, we prove that a homotopy classification of vector bundles of rank $n$ over rigid analytic quasi-Stein spaces follows from $\mathbb{A}^1_\mathrm{rig}$-homotopy invariance of vector bundles. This $\mathbb{A}^1_\mathrm{rig}$-homotopy invariance is equivalent to a rigid analytic version of Lindel's solution to the Bass--Quillen conjecture. Moreover, we establish a homotopy classification of line bundles over rigid analytic quasi-Stein spaces. In fact, line bundles are classified by infinite projective space.