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arXiv · 2505.04760

Isotropic Torsors on Smooth Algebras over Pr\"ufer Rings

Arnab Kundu
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Abstract

The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group over a regular semilocal ring is itself trivial. Extending the work of \v{C}esnavi\v{c}ius and Fedorov, we prove a non-noetherian analogue of this conjecture for rings $A$ that are semilocalisations of smooth schemes over valuation rings of rank one, and for reductive $A$-group schemes $G$ that are totally isotropic. Roughly speaking, such group schemes are characterised by the existence of a parabolic subgroup of their adjoint quotients. Since quasi-split groups are totally isotropic, our result, in particular, generalises the Grothendieck--Serre result of Guo--Liu and the author's thesis. Our proof relies on a new instance of Gabber's presentation lemma, obtained by extending techniques developed in the author's thesis.

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Arnab Kundu. 2025-05-07. Isotropic Torsors on Smooth Algebras over Pr\"ufer Rings. https://arxiv.org/abs/2505.04760

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