Grothendieck-Serre in the quasi-split unramified case

TL;DR

This paper proves the Grothendieck-Serre conjecture for quasi-split reductive groups over unramified regular local rings, using advanced techniques like Noether normalization and a presentation lemma for mixed characteristic.

Contribution

It establishes the conjecture in the unramified case for quasi-split groups, extending previous results and introducing new methods for mixed characteristic scenarios.

Findings

Proved the Grothendieck-Serre conjecture for quasi-split groups over unramified rings.

Developed a version of Noether normalization over discrete valuation rings.

Created a presentation lemma for smooth relative curves in mixed characteristic.

Abstract

The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme $G$ over a regular local ring $R$ is trivial. We settle it in the case when $G$ is quasi-split and $R$ is unramified. Some of the techniques that allow us to overcome obstacles that have so far kept the mixed characteristic case out of reach include a version of Noether normalization over discrete valuation rings, as well as a suitable presentation lemma for smooth relative curves in mixed characteristic that facilitates passage to the relative affine line via excision and patching.