On the Grothendieck--Serre conjecture for projective smooth schemes over a DVR

Abstract

The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme $G$ over a regular local ring $R$ is trivial. The mixed characteristic case of the conjecture is widely open. We consider the following setup. Let $A$ be a mixed characteristic DVR, $G$ a reductive group scheme over $A$, $X$ an irreducible smooth projective $A$-scheme, $\mathcal{G}$ a principal $G$-bundle over $X$. Suppose $\mathcal{G}$ is generically trivial. We prove that in this case $\mathcal{G}$ is Zariski locally trivial. This result confirms the conjecture.

11 pages. arXiv admin note: text overlap with arXiv:1707.01767