A step towards the mixed-characteristic Grothendieck–Serre conjecture
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A. Tsybyshev
Translated by: A. Tsybyshev - St. Petersburg Math. J. 31 (2020), 181-187
- 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.References
- Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Lecture Notes in Mathematics, Vol. 152, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274459
- Mingchu Gao, Mild integrated $C$-existence families and abstract Cauchy problems, Northeast. Math. J. 14 (1998), no. 1, 95–104. MR 1633911
- A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. 20 (1964), 259 (French). MR 173675
- D. Mumford, J. Fogarty, and F. Kirwan, Geometric invariant theory, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906
- R. W. Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. in Math. 65 (1987), no. 1, 16–34. MR 893468, DOI 10.1016/0001-8708(87)90016-8
- Jean-Louis Colliot-Thélène and Manuel Ojanguren, Espaces principaux homogènes localement triviaux, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 97–122 (French). MR 1179077
- Alexander Grothendieck, Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats, Séminaire Bourbaki, Vol. 5, Soc. Math. France, Paris, 1995, pp. Exp. No. 190, 299–327 (French). MR 1603475
- Yevsey A. Nisnevich, Espaces homogènes principaux rationnellement triviaux et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 1, 5–8 (French, with English summary). MR 756297
- Yevsey Nisnevich, Rationally trivial principal homogeneous spaces, purity and arithmetic of reductive group schemes over extensions of two-dimensional regular local rings, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), no. 10, 651–655 (English, with French summary). MR 1054270
- M. S. Raghunathan, Principal bundles admitting a rational section, Invent. Math. 116 (1994), no. 1-3, 409–423. MR 1253199, DOI 10.1007/BF01231567
- M. S. Raghunathan, Erratum: “Principal bundles admitting a rational section” [Invent. Math. 116 (1994), no. 1-3, 409–423; MR1253199 (95f:14093)], Invent. Math. 121 (1995), no. 1, 223. MR 1345290, DOI 10.1007/BF01884296
- I. Panin, A. Stavrova, and N. Vavilov, On Grothendieck-Serre’s conjecture concerning principal $G$-bundles over reductive group schemes: I, Compos. Math. 151 (2015), no. 3, 535–567. MR 3320571, DOI 10.1112/S0010437X14007635
- I. Panin, Proof of Grothendieck–Serre conjecture on principal $G$-bundles over regular local rings containing a finite field, arXiv:1406.0247.
- Roman Fedorov and Ivan Panin, A proof of the Grothendieck-Serre conjecture on principal bundles over regular local rings containing infinite fields, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 169–193. MR 3415067, DOI 10.1007/s10240-015-0075-z
Bibliographic 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
- 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
- © Copyright 2019 American Mathematical Society
- Journal: St. Petersburg Math. J. 31 (2020), 181-187
- MSC (2010): Primary 14L15
- DOI: https://doi.org/10.1090/spmj/1591
- MathSciNet review: 3932824