A presentation theorem for smooth projective schemes over discrete valuation rings

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 ${\mathbb{A}}_U^1$ for a semilocal affine scheme $U$.