Unramified Grothendieck-Serre for simply-connected group schemes satisfying an isotropy condition via unipotent chains

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 over $R$ 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 quasi-split 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 the spectrum of $R$, there is a closed subset $Z$ of the spectrum of $R$ of codimension at least two such the torsor trivializes over every affine scheme that factors through $U-Z$.