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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.

The Bass–Quillen Conjecture for Torsors Over Valuation Rings