The Bass--Quillen conjecture for torsors over valuation rings
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912365686030336 |
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| author | Guo, Ning Liu, Fei |
| author_facet | Guo, Ning Liu, Fei |
| contents | 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 $\mathbb{A}^N_A$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10163 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Bass--Quillen conjecture for torsors over valuation rings Guo, Ning Liu, Fei Algebraic Geometry Commutative Algebra 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 $\mathbb{A}^N_A$ 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. |
| title | The Bass--Quillen conjecture for torsors over valuation rings |
| topic | Algebraic Geometry Commutative Algebra |
| url | https://arxiv.org/abs/2503.10163 |