On the Grothendieck-Serre conjecture on principal bundles in mixed characteristic

TL;DR

This paper proves the Grothendieck-Serre conjecture for a broad class of regular local rings in mixed characteristic, specifically when the group scheme is split, extending known results beyond rings containing fields.

Contribution

It establishes the conjecture for regular local rings not containing fields in the case of split reductive group schemes, advancing understanding in mixed characteristic cases.

Findings

Proves the Grothendieck-Serre conjecture for certain regular local rings in mixed characteristic.

Extends known results to rings not containing fields.

Focuses on split reductive group schemes.

Abstract

Let R be a regular local ring. Let G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle over R is trivial if it is trivial over the quotient field of R. The conjecture is known when R contains a field. We prove the conjecture for a large class of regular local rings not containing fields in the case when G is split.