Torsors on the complement of a smooth divisor
Kestutis Cesnavicius
TL;DR
This work completes the Nisnevich conjecture in equal characteristic and extends to several mixed-characteristic DVR contexts under total isotropy or quasi-split descent, by reducing torsor triviality questions to relative-curve settings and applying a robust Grothendieck–Serre framework. Central to the approach are Gabber–Quillen presentation lemmas, excision, and patching dévissages, together with a novel finite-field trick to bypass obstructions, enabling extension and triviality results for reductive G-torsors on complements of smooth divisors. The authors also provide a new equal-characteristic proof of the Bass–Quillen conjecture for reductive torsors over affine spaces, linking Nisnevich phenomena to classical descent questions. The methodology separates base-independent, “axiomatic” torsor behavior on relative curves from the base geometry, yielding broad applicability to equal/mixed characteristic cases and to generalized descent problems for torsors under reductive groups.
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 dévissages 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.