论文标题
平滑除数的补充上的扭矩
Torsors on the complement of a smooth divisor
论文作者
论文摘要
我们以平等的特征完成了Nisnevich猜想的证明：对于平稳的代数品种$ x $，$ k $，a $ k $ -smooth divisor $ d \ d \ subset x $，以及还原的$ x $ x $ -group $ g $，其基础$ g_d $完全是$ g_d $是$ g_d $ ns n is iStriv $ g $ g $ - Zariski在$ x $上半落。在混合特征中，当$ k $被离散的估值戒指$ o $替换时，我们会显示同样的情况，除数$ d $是$ x $ $ x $的封闭$ o $ $ - $ g $是quasi-split或$ g $仅定义了$ x \ setminus d $，但下降了$ split $ a $ a。 Kisin-Pappas型变体）。我们的论点结合了Gabber-Quillen风格的演示引理和切除和重新组装Dévissages，以减少在相对仿射线上分析一般琐碎的Torsors。作为此分析的副产品，我们为低于$ \ mathbb {a}^d_r $的低音Quillen猜想提供了新的证明。
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.
