On the Gille theorem for the relative projective line

Abstract

Let $X$ be a Noetherian separated scheme. Let $G$ be a reductive $X$-group scheme, and let $E$ be a principal $G$-bundle over ${\mathbb{P}}_X^1$. We prove that if the restriction of $E$ to $\infty \times X$ is Zariski locally trivial, then $E$ is itself Zariski locally trivial.