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arXiv · 2503.10163

The Bass--Quillen conjecture for torsors over valuation rings

Ning Guo·Fei Liu
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Abstract

For a valuation ring $V$, a smooth $V$-algebra $A$, and a reductive $V$-group scheme $G$ satisfying a certain natural isotropicity condition, we prove that every Nisnevich $G$-torsor on $\mathbb{A}^N_A$ descends to a $G$-torsor on $A$. As a corollary, we generalize Raghunathan's theorem on torsors over affine spaces to a relative setting. We also extend several affine representability results of Asok, Hoyois, and Wendt from equi-characteristics to mixed characteristics. Our proof relies on previous work on the purity of reductive torsors over smooth relative curves and the Grothendieck--Serre conjecture for constant reductive group schemes.

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Ning Guo, Fei Liu. 2025-03-13. The Bass--Quillen conjecture for torsors over valuation rings. https://arxiv.org/abs/2503.10163

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