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arXiv · 2302.02842

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

Ning Guo·Ivan Panin
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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.

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Ning Guo, Ivan Panin. 2023-02-06. On the Grothendieck--Serre conjecture for projective smooth schemes over a DVR. https://arxiv.org/abs/2302.02842

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