The Bass--Quillen conjecture for torsors over valuation rings
Ning Guo, Fei Liu
TL;DR
The paper proves a Bass–Quillen-type descent for torsors under totally isotropic reductive group schemes over rings that are ind-smooth over Prüfer bases, establishing a bijection $H^1_{ m Nis}(A,G) \cong H^1_{ m Nis}(\mathbb{A}^N_A,G)$ and thereby generalizing Raghunathan’s theorem to a relative, mixed-characteristic setting. The authors develop a robust framework combining Quillen patching, Gabber-style inverse patching, and a Lindel-type geometric presentation lemma, and they leverage purity for reductive torsors and the Grothendieck–Serre conjecture for constant reductive groups to obtain the descent results. They also extend affine representability results to mixed characteristics, showing $H^1_{ m Nis}(U,G) \simeq [U,\mathbf{B}G]_{\mathbb{A}^1}$ for $U$ smooth affine over a Prüfer base and proving $G/H$ is $\mathbb{A}^1$-naive in suitable settings. These results bridge equi- and mixed-characteristic contexts and deepen the interaction between torsor theory and $\mathbb{A}^1$-homotopy in arithmetic geometry.
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.