Torsors on the complement of a smooth divisor
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911305403727872 |
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| author | Cesnavicius, Kestutis |
| author_facet | Cesnavicius, Kestutis |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_08233 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Torsors on the complement of a smooth divisor Cesnavicius, Kestutis Algebraic Geometry Number Theory Primary 14L15, Secondary 14M17, 20G10 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. |
| title | Torsors on the complement of a smooth divisor |
| topic | Algebraic Geometry Number Theory Primary 14L15, Secondary 14M17, 20G10 |
| url | https://arxiv.org/abs/2204.08233 |