Towards a non-archimedean analytic analog of the Bass-Quillen conjecture

TL;DR

This paper proposes a non-archimedean analytic analog of the Bass-Quillen conjecture, proving it for the Picard group in rank-1 cases and for vector bundles over certain regular affinoid algebras.

Contribution

It introduces a new non-archimedean analog of the Bass-Quillen conjecture and proves it in specific cases involving Picard groups and vector bundles.

Findings

Proved the conjecture for the Picard group in rank-1 cases.

Extended results to Grothendieck groups of vector bundles over certain affinoid algebras.

Established the conjecture for regular affinoid algebras with regular models.

Abstract

We suggest an analog of the Bass-Quillen conjecture for smooth affinoid algebras over a complete non-archimedean field. We prove this in the rank-1 case, i.e. for the Picard group. For complete discretely valued fields and regular affinoid algebras that admit a regular model (automatic if the residue characteristic is zero) we prove a similar statement for the Grothendieck group of vector bundles.