A step towards the mixed-characteristic Grothendieck–Serre conjecture

摘要

In the present paper, the objects of consideration include a regular semilocal Noetherian scheme $ W$, a reductive group scheme $ G$ over $ W$ and a principal $ G$-bundle over $ mathbb{P}^1_W$. The main theorem of the paper states that if the restriction of such a $ G$-bundle to each closed fiber is trivial, then the original bundle is an inverse image of some principal $ G$-bundle on $ W$. For the case when the scheme $ W$ is equicharacteristic, this theorem was proved in a paper by Panin, Stavrova, and Vavilov on the Grothendieck-Serre conjecture. That equicharacteristic case of the theorem was used in a paper by Fedorov and Panin, and in another paper by Panin, to prove the Grothendieck-Serre conjecture itself in the equicharacteristic case. The main theorem of the present paper may be useful for proving the general case of the Grothendieck-Serre conjecture.