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 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.