The Bass--Quillen conjecture for torsors over valuation rings

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 ${\mathbb{A}}_A^N$ 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.