On the generalized Bass--Quillen conjecture in dimension 2

TL;DR

This paper proves that over regular rings of dimension at most 2, Zariski-locally trivial principal G-bundles over polynomial extensions are extended from the base ring, generalizing a key case of the Bass--Quillen conjecture.

Contribution

It extends the dimension 2 case of the Bass--Quillen conjecture to split reductive groups over regular rings of dimension ≤ 2.

Findings

Zariski-locally trivial principal G-bundles over A[x_1,...,x_n] are extended from A

Generalization of the Bass--Quillen conjecture to split reductive groups in dimension 2

Supports the conjecture for regular rings of low dimension

Abstract

Let $A$ be a regular ring of dimension $\le 2$. Let $G$ be a reductive group over $A$ such that its derived group is a split, i.e. a Chevalley--Demazure, semisimple group. We prove that every Zariski-locally trivial principal $G$-bundle over $A[x_1,\ldots,x_n]$ is extended from $A$, for any $n\ge 1$. This result generalizes to split reductive groups the dimension $2$ case of the Bass--Quillen conjecture on finitely generated projective modules, settled in positive by M. P. Murthy.