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A step towards the mixed-characteristic Grothendieck-Serre conjecture


Author: A. Tsybyshev
Translated by: A. Tsybyshev
Original publication: Algebra i Analiz, tom 31 (2019), nomer 1.
Journal: St. Petersburg Math. J. 31 (2020), 181-187
MSC (2010): Primary 14L15
DOI: https://doi.org/10.1090/spmj/1591
Published electronically: December 3, 2019
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Abstract: In the present paper, the objects of consideration include a regular semilocal Noetherian scheme $ W$, a reductive group scheme $ G$ over $ W$ and a principal $ G$-bundle over $ \mathbb{P}^1_W$. The main theorem of the paper states that if the restriction of such a $ G$-bundle to each closed fiber is trivial, then the original bundle is an inverse image of some principal $ G$-bundle on $ W$. For the case when the scheme $ W$ is equicharacteristic, this theorem was proved in a paper by Panin, Stavrova, and Vavilov on the Grothendieck-Serre conjecture. That equicharacteristic case of the theorem was used in a paper by Fedorov and Panin, and in another paper by Panin, to prove the Grothendieck-Serre conjecture itself in the equicharacteristic case. The main theorem of the present paper may be useful for proving the general case of the Grothendieck-Serre conjecture.


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Additional Information

A. Tsybyshev
Affiliation: St. Petersburg Branch, V. A. Steklov Mathematical Institute, Russian Academy of Sciences and Euler International Mathematical Institute of the Russian Academy of Sciences, Fontanka 27 , St. Petersburg, Russia; Chebyshev Laboratory, St. Petersburg State University, 14th Line 29B, Vasilyevsky Island, St. Petersburg 199178, Russia
Email: emperortsy@gmail.com

DOI: https://doi.org/10.1090/spmj/1591
Keywords: Reductive group schemes, pricipal bundles, Grothendieck--Serre conjecture, mixed characteristic, quotient sheaves
Received by editor(s): April 24, 2019
Published electronically: December 3, 2019
Additional Notes: The author thanks the Presidium of RAS program no. 01 “Basic mathematics and its applications” (grant PRAS-18-01) for support
Article copyright: © Copyright 2019 American Mathematical Society