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 Zap. Nauchn. Sem. POMI, 2019, Volume 479, Pages 131–136 (Mi znsl6754)

A short proof of a theorem due to O. Gabber

I. A. Panin

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

Abstract: A very short proof of an unpublished result due to O. Gabber is given. More presicely, let $R$ be a regular local ring, containing a finite field $k$. Let $\mathbf{G}$ be a simply-connected reductive group scheme over $k$. We prove that a principal $\mathbf{G}$-bundle over $R$ is trivial, if it is trivial over the fraction field of $R$. This is the mentioned unpublished result due to O. Gabber. We derive this result from a purely geometric one proven in another paper of the author and stated in the Introduction.

Key words and phrases: semi-simple algebraic group, principal bundle, Grothendieck–Serre conjecture.

 Funding Agency Grant Number Russian Foundation for Basic Research 19-01-00513 The author acknowledges support of the RFBR grant No. 19-01-00513.

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Citation: I. A. Panin, “A short proof of a theorem due to O. Gabber”, Algebra and number theory. Part 2, Zap. Nauchn. Sem. POMI, 479, POMI, St. Petersburg, 2019, 131–136

Citation in format AMSBIB
\Bibitem{Pan19} \by I.~A.~Panin \paper A short proof of a theorem due to O.~Gabber \inbook Algebra and number theory. Part~2 \serial Zap. Nauchn. Sem. POMI \yr 2019 \vol 479 \pages 131--136 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6754}