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

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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 setsinduced by the inclusion of $R$ in $K$ has trivial kernel. This result was proved in [1] for regularlocal rings $R$ containing an infinite field.

About the authors

Ivan Alexandrovich Panin

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

Email: paniniv@gmail.com
Doctor of physico-mathematical sciences

References

  1. R. Fedorov, I. Panin, “A proof of the Grothendieck–Serre conjecture on principal bundles over regular local ring containing infinite fields”, Publ. Math. Inst. Hautes Etudes Sci., 122 (2015), 169–193
  2. Schemas en groupes, Seminaire de geometrie algebrique du Bois Marie 1962/64 (SGA 3), v. I, Lecture Notes in Math., 151, eds. M. Demazure, A. Grothendieck, Springer-Verlag, Berlin–New York, 1970, xv+564 pp.
  3. J. P. Serre, “Les espaces fibres algebriques”, Anneaux de Chow et applications, Seminaire C. Chevalley; 2e annee, Secretariat mathematique, Paris, 1958, Exp. No. 1, 37 pp.
  4. A. Grothendieck, Seminaire Chevalley; 2-e annee, Anneaux de Chow et applications, Secretariat mathematique, Paris, 1958, Exp. No. 5, 29 pp.
  5. A. Grothendieck, “Le group de Brauer. II. Theorie cohomologique”, Dix exposes sur la cohomologie de schemas, Adv. Stud. Pure Math., 3, North-Holland, Amsterdam, 1968, 67–87
  6. I. Panin, “On Grothendieck–Serre conjecture concerning principal bundles”, Proceedings of the international congress of mathematicians (Rio de Janeiro, 2018), v. 2, World Sci. Publ., Hackensack, NJ, 2018, 201–221
  7. И. А. Панин, “Совершенные тройки и гомотопии отображений мотивных пространств”, Изв. РАН. Сер. матем., 83:4 (2019), 158–193
  8. I. Panin, “Nice triples and the Grothendieck–Serre conjecture concerning principal $G$-bundles over reductive group schemes”, Duke Math. J., 168:2 (2019), 351–375
  9. И. А. Панин, “Две теоремы чистоты и гипотеза Гротендика–Серра о главных $mathbf G$-расслоениях”, Матем. сб., 211:12 (2020), 123–142
  10. B. Poonen, “Bertini theorems over finite fields”, Ann. of Math. (2), 160:3 (2004), 1099–1127
  11. F. Charles, B. Poonen, “Bertini irreducibility theorems over finite fields”, J. Amer. Math. Soc., 29:1 (2016), 81–94
  12. M. Ojanguren, I. Panin, “A purity theorem for the Witt group”, Ann. Sci. Ecole Norm. Sup. (4), 32:1 (1999), 71–86
  13. M. Ojanguren, I. Panin, “Rationally trivial hermitian spaces are locally trivial”, Math. Z., 237:1 (2001), 181–198
  14. J.-L. Colliot-Thelène, M. Ojanguren, “Espaces principaux homogènes localement triviaux”, Inst. Hautes Etudes Sci. Publ. Math., 75:2 (1992), 97–122
  15. P. Gille, “Torseurs sur la droite affine”, Transform. Groups, 7:3 (2002), 231–245
  16. Ph. Gille, “Le problème de Kneser–Tits”, Seminaire N. Bourbaki, v. 2007/2008, Asterisque, 326, Soc. Math. France, Paris, 2009, Exp. No. 983, vii, 39–81
  17. I. Panin, A. Stavrova, N. Vavilov, “On Grothendieck–Serre's conjecture concerning principal $G$-bundles over reductive group schemes: I”, Compos. Math., 151:3 (2015), 535–567

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