On the Gille theorem for the relative projective line

TL;DR

For a Noetherian separated base X and a reductive X-group scheme G, the paper proves that a principal G-bundle 𝒢 over the relative projective line 𝔓^1_X with 𝒢|_{∞×X} Zariski locally trivial is itself Zariski locally trivial on 𝔓^1_X. The authors develop a two-pronged reduction to torus and isotropic simply connected cases via a two-pullbacks framework and endomorphisms of 𝔓^1_U, then implement patching with finite étale centers Y to obtain a detailed classification and extension results. They obtain corollaries for the torus and simply connected semisimple cases, extend the results to general reductive groups, and supply an Appendix with Bertini-type arguments to produce the required finite étale subschemes. These results connect to the Grothendieck–Serre conjecture in a relative setting and provide structural tools for principal bundles over relative projective lines, including a description in terms of double cosets and patching data. Overall, the work advances understanding of when generically trivial principal bundles become locally trivial after base change, with implications for Serre-type questions in mixed characteristic.

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 $\mathbb{P}^1_X$. We prove that if the restriction of $E$ to $\infty\times X$ is Zariski locally trivial, then $E$ is itself Zariski locally trivial.

Theorems & Definitions (71)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Lemma 2.1
• proof
• Theorem 2.2