Two purity theorems and the Grothendieck-Serre conjecture concerning principal $\mathbf G$-bundles

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Abstract

The main results of the paper are two purity theorems for reductive group schemes over regular local rings containing a field. Using these two theorems a well-known Grothendieck-Serre conjecture on principal bundles is reduced to the simply-connected case. We point out that the mentioned reduction is one of the major steps in the proof of the conjecture that the author published in another work. Bibliography: 25 titles.

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. J.-L. Colliot-Thelène, M. Ojanguren, “Espaces principaux homogènes localement triviaux”, Inst. Hautes Etudes Sci. Publ. Math., 75:2 (1992), 97–122
  2. J.-L. Colliot-Thelène, R. Parimala, R. Sridharan, “Un theorème de purete locale”, C. R. Acad. Sci. Paris Ser. I Math., 309:14 (1989), 857–862
  3. Schemas en groupes, Seminaire de geometrie algebrique du Bois Marie 1962/64 (SGA 3), dirige par M. Demazure, A. Grothendieck, v. III, Lecture Notes in Math., 153, Structure des schemas en groupes reductifs, Rev. reprint, Springer-Verlag, Berlin–New York, 1970, viii+529 pp.
  4. D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Grad. Texts in Math., 150, Springer-Verlag, New York, 1995, xvi+785 pp.
  5. R. Fedorov, I. Panin, “A proof of Grothendieck–Serre conjecture on principal bundles over regular local rings containing infinite fields”, Publ. Math. Inst. Hautes Etudes Sci., 122 (2015), 169–193
  6. A. Grothendieck, “Torsion homologique et sections rationnalles”, Seminaire C. Chevalley (2e annee), v. 3, Anneaux de Chow et applications, Secretariat mathematique, Paris, 1958, Exp. No. 5, 29 pp.
  7. 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
  8. N. Guo, “The Grothendieck–Serre conjecture over semilocal Dedekind rings”, Transform. Groups, Publ. online: 2020, 1–21
  9. Y. A. Nisnevich, “Espaces homogènes principaux rationnellement triviaux et arithmetique des schemas en groupes reductifs sur les anneaux de Dedekind [Rationally trivial principal homogeneous spaces and arithmetic of reductive group schemes over Dedekind rings]”, C. R. Acad. Sci. Paris Ser. I Math., 299:1 (1984), 5–8
  10. M. Ojanguren, I. Panin, “A purity theorem for the Witt group”, Ann. Sci. Ecole Norm. Sup. (4), 32:1 (1999), 71–86
  11. M. Ojanguren, I. Panin, “Rationally trivial Hermitian spaces are locally trivial”, Math. Z., 237:1 (2001), 181–198
  12. 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
  13. I. A. Panin, “On Grothendieck–Serre's conjecture concerning principal $G$-bundles over reductive group schemes: II”, Изв. РАН. Сер. матем., 80:4 (2016), 131–162
  14. 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
  15. И. А. Панин, “Совершенные тройки и гомотопии отображений мотивных пространств”, Изв. РАН. Сер. матем., 83:4 (2019), 158–193
  16. 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
  17. И. А. Панин, “Доказательство гипотезы Гротендика–Серра о главных расслоениях над регулярным локальным кольцом, содержащим поле”, Изв. РАН. Сер. матем., 84:4 (2020), 169–186
  18. И. А. Панин, А. А. Суслин, “Об одной гипотезе Гротендика, касающейся алгебр Адзумайа”, Алгебра и анализ, 9:4 (1997), 215–223
  19. B. Poonen, “Bertini theorems over finite fields”, Ann. of Math. (2), 160:3 (2004), 1099–1127
  20. F. Charles, B. Poonen, “Bertini irreducibility theorems over finite fields”, J. Amer. Math. Soc., 29:1 (2016), 81–94
  21. M. Rost, “Durch Normengruppen definierte birationale Invarianten”, C. R. Acad. Sci. Paris Ser. I Math., 310:4 (1990), 189–192
  22. J.-P. Serre, “Espaces fibres algebriques”, Seminaire C. Chevalley (2e annee), v. 3, Anneaux de Chow et applications, Secretariat mathematique, Paris, 1958, Exp. No. 1, 37 pp.
  23. A. Suslin, V. Voevodsky, “Singular homology of abstract algebraic varieties”, Invent. Math., 123:1 (1996), 61–94
  24. V. Voevodsky, “Cohomological theory of presheaves with transfers”, Cycles, transfers, and motivic homology theories, Ann. of Math. Stud., 143, Princeton Univ. Press, Princeton, NJ, 2000, 87–137
  25. К. В. Зайнуллин, “Гипотеза Гротендика о главных однородных пространствах для некоторых классических алгебраических групп”, Алгебра и анализ, 12:1 (2000), 150–184

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