On the Grothendieck–Serre conjecture on principal bundles in mixed characteristic

Fedorov, Roman

doi:10.1090/tran/8490

On the Grothendieck–Serre conjecture on principal bundles in mixed characteristic
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Trans. Amer. Math. Soc. 375 (2022), 559-586 Request permission

Abstract:

Let $R$ be a regular local ring. Let $\mathbf {G}$ be a reductive $R$-group scheme. A conjecture of Grothendieck and Serre predicts that a principal ${\mathbf {G}}$-bundle over $R$ is trivial if it is trivial over the quotient field of $R$. The conjecture is known when $R$ contains a field. We prove the conjecture for a large class of regular local rings not containing fields in the case when $\mathbf {G}$ is split.

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