Torsors on the complement of a smooth divisor

Abstract: We complete the proof of the Nisnevich conjecture in equal characteristic: for a smooth algebraic variety $X$ over a field $k$, a $k$-smooth divisor $D \subset X$, and a reductive $X$-group $G$ whose base change $G_D$ is totally isotropic, we show that each generically trivial $G$-torsor on $X\setminus D$ trivializes Zariski semilocally on $X$. In mixed characteristic, we show the same when $k$ is a replaced by a discrete valuation ring $O$, the divisor $D$ is the closed $O$-fiber of $X$, and either $G$ is quasi-split or $G$ is only defined over $X \setminus D$ but descends to a quasi-split group over $\mathrm{Frac}(O)$ (a Kisin-Pappas type variant). Our arguments combine Gabber-Quillen style presentation lemmas with excision and reembedding d\'{e}vissages to reduce to analyzing generically trivial torsors over a relative affine line. As a byproduct of this analysis, we give a new proof for the Bass-Quillen conjecture for reductive group torsors over $\mathbb{A}^d_R$ in equal characteristic.