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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya // Archive

Izv. RAN. Ser. Mat., 2020 Volume 84, Issue 4, Pages 169–186 (Mi im8982)

This article is cited in 12 papers

Proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing a field

I. A. Panin

St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: Let $R$ be a regular local ring containing a field. Let $\mathbf{G}$ be a reductive group scheme over $R$. We prove that a principal $\mathbf{G}$-bundle over $R$ is trivial if it is trivial over the field of fractions of $R$. In other words, if $K$ is the field of fractions of $R$, then the map
$$ H^1_{\mathrm{et}}(R,\mathbf{G})\to H^1_{\mathrm{et}}(K,\mathbf{G}) $$
of the non-Abelian cohomology pointed sets induced by the inclusion of $R$ in $K$ has trivial kernel. This result was proved in [1] for regular local rings $R$ containing an infinite field.

Keywords: reductive group schemes, principal bundles, Grothendieck–Serre conjecture.

UDC: 512.74+512.723

MSC: 14L10, 20G10, 20G35

Received: 18.10.2019
Revised: 31.01.2020

DOI: 10.4213/im8982


 English version:
Izvestiya: Mathematics, 2020, 84:4, 780–795

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© Steklov Math. Inst. of RAS, 2025