Isotropic Torsors on Smooth Algebras over Pr\"ufer Rings

TL;DR

This paper proves a non-noetherian analogue of the Grothendieck--Serre conjecture for certain smooth algebras over valuation rings, extending previous results to a broader class of rings and group schemes.

Contribution

It extends the Grothendieck--Serre conjecture to semilocalizations of smooth schemes over valuation rings of rank one for totally isotropic reductive group schemes.

Findings

Proves triviality of torsors in new non-noetherian setting

Generalizes previous results to broader class of rings and groups

Introduces a new instance of Gabber's presentation lemma

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.