On the generalized Bass--Quillen conjecture in dimension 2

TL;DR

This work proves the generalized Bass--Quillen conjecture in dimension at most two for reductive groups with split derived group: every Zariski-locally trivial principal G-bundle over $A[x_1,\dots,x_n]$ is extended from A for all $n\ge1$ when A is regular of dim ≤2. The authors reduce to the simply connected split case, employing patching, descent, and stability techniques to establish vanishing results in two variables and then extend to multiple variables via induction and the AHW local-global principle. The results extend Murthy’s dimension-two GL_n case to all split reductive groups, reinforcing the local-global paradigm for projective-like objects over polynomial rings in low dimension. Overall, the paper advances understanding of how isotropy and split-ness interact with Bass–Quillen-type phenomena in small dimensions.

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.

Theorems & Definitions (20)

• Conjecture 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Corollary 2.2
• proof
• Theorem 2.3
• proof
• Lemma 2.4
• Theorem 3.1