A presentation theorem for smooth projective schemes over discrete valuation rings

TL;DR

This paper proves a geometric presentation theorem for smooth projective schemes over discrete valuation rings and applies it to principal G-bundles, showing their triviality under certain conditions.

Contribution

It introduces a new presentation theorem for schemes over DVRs and demonstrates its application to the triviality of principal G-bundles over such schemes.

Findings

Proved a geometric presentation theorem for smooth projective schemes over DVRs.

Showed that certain principal G-bundles are trivial over these schemes.

Extended results to bundles over affine lines for semilocal schemes.

Abstract

In this article, we give a proof for a geometric presentation theorem for any irreducible scheme $X$ smooth projective over a discrete valuation ring $R$. As a consequence, for any reductive $R$-group scheme $\mathbf{G}$, we prove that any generically trivial principal $\mathbf{G}$-bundle over $X$ glued to a principal $\mathbf{G}_U$-bundle over the affine line $\mathbb{A}^1_U$ for a semilocal affine scheme $U$.