Unramified case of Grothendieck-Serre for totally isotropic simply-connected group schemes

TL;DR

This paper proves a case of the Grothendieck-Serre conjecture for certain group schemes over semilocal rings, showing torsor triviality under specific conditions and simplifying proofs in unramified cases.

Contribution

It establishes new cases of the Grothendieck-Serre conjecture for simply-connected reductive group schemes and introduces the concept of unipotent chains of torsors.

Findings

Proves triviality of G-torsors over rings with regular fibers.

Simplifies proof of the conjecture in unramified cases.

Shows torsors trivialize outside codimension two subsets.

Abstract

We prove a case of the Grothendieck-Serre conjecture: let $R$ be a Noetherian semilocal flat algebra over a Dedekind domain such that all fibers of $R$ are geometrically regular; let $G$ be a simply-connected reductive $R$-group scheme having a strictly proper parabolic subgroup scheme. Then a $G$-torsor is trivial, provided that it is trivial over the total ring of fractions of $R$. We also simplify the proof of the conjecture in the quasisplit unramified case. The argument is based on the notion of a unipotent chain of torsors that we introduce. We also prove that if $R$ is a Noetherian normal domain and $G$ is as above, then for any generically trivial torsor over an open subset $U$ of Spec $R$, there is a closed subset $Z$ of Spec $R$ of codimension at least two such the torsor trivializes over any affine open of $U-Z$.