On the Grothendieck--Serre conjecture for projective smooth schemes over a DVR

TL;DR

This paper proves that for smooth projective schemes over a mixed characteristic DVR, any generically trivial principal G-bundle is actually trivial locally, confirming a special case of the Grothendieck--Serre conjecture.

Contribution

It establishes the Zariski local triviality of generically trivial principal G-bundles over smooth projective schemes in mixed characteristic, confirming a significant case of the conjecture.

Findings

Generically trivial G-bundles are Zariski locally trivial over smooth projective schemes.

Confirms the Grothendieck--Serre conjecture in mixed characteristic case.

Advances understanding of torsors over regular local rings in algebraic geometry.

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. The mixed characteristic case of the conjecture is widely open. We consider the following setup. Let $A$ be a mixed characteristic DVR, $G$ a reductive group scheme over $A$, $X$ an irreducible smooth projective $A$-scheme, $\mathcal G$ a principal $G$-bundle over $X$. Suppose $\mathcal G$ is generically trivial. We prove that in this case $\mathcal G$ is Zariski locally trivial. This result confirms the conjecture.