Notes on a Grothendieck–Serre conjecture in mixed characteristic case

I. Panin

Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 35, Tome 484 (2019), pp. 138-148

Résumé

Let $R$ be a discrete valuation ring with an infinite residue field, $X$ be a smooth projective curve over $R$. Let $\mathbf{G}$ be a simple simply-connected group scheme over $R$ and $E$ be a principal $\mathbf{G}$-bundle over $X$. We prove that $E$ is trivial locally for the Zariski topology on $X$ providing it is trivial over the generic point of $X$. The main aim of the present paper is to develop a method rather than to get a very strong concrete result.

Bibliographie

[1] J.-L. Colliot-Thélène, M. Ojanguren, “Espaces principaux homogènes localement triviaux”, Publ. Math. IHÉS, 75:2 (1992), 97–122 | DOI | MR | Zbl

[2] M. Demazure, A. Grothendieck, Schémas en groupes, Lect. Notes Math., 151–153, Springer-Verlag, Berlin–Heidelberg–New York, 1970 | MR

[3] M. Demazure, A. Grothendieck, Structure des schémas en groupes réductifs, Lect. Notes Math., 153 | MR

[4] D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995 | DOI | MR | Zbl

[5] R. Fedorov, I. Panin, “A proof of Grothendieck–Serre conjecture on principal bundles over a semilocal regular ring containing an infinite field”, Publ. Math. IHE'S, 122 (2015), 169–193 ; arXiv: 1211.2678v2 | DOI | MR | Zbl

[6] Ph. Gille, “Torseurs sur la droite affine”, Transform. Groups, 7:3 (2002), 231–245 | DOI | MR | Zbl

[7] Ph. Gille, “Le problème de Kneser-Tits”, Astérisque, 326, no. 983, 2010, 39–81 ; Séminaire Bourbaki, 2007/2008, 2009 | MR | MR

[8] A. Grothendieck, Torsion homologique et sections rationnelles, Anneaux de Chow et applications, Séminaire Chevalley, 2-e année, Secrétariat mathématique, Paris, 1958 | MR

[9] A. Grothendieck, “Le group de Brauer II”, Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968 | MR | Zbl

[10] A. Grothendieck, “Le group de Brauer III: Exemples et compléments”, Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968 | MR | Zbl

[11] A. Grothendieck, “Technique de descente et théorèmes d'existence en géométrie algébrique: I. Generalites”, Descente par morphismes fidèlement plats, Seminaire Bourbaki, 5, no. 190, Soc. Math. France, Paris, 1995, 299–327 | MR

[12] A. Grothendieck, “Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : III. Étude cohomologie des faisceaux cohérents, Première partie”, Publ. Math. IHÉS, 11, 1961, 5–167 | DOI | MR

[13] A. Grothendieck, “Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné) : IV. Étude locale des schémas et des morphismes de schémas, Seconde partie”, Publ. Math. IHÉS, 24, 1965, 5–231 | DOI | MR

[14] A. Grothendieck, “Le groupe de Brauer II”, Dix exposés sur la cohomologie des schémas, North-Holland, Amsterdam, 1968 | MR | Zbl

[15] F. Morel, V. Voevodsky, “$A^1$-homotopy theory of schemes”, Publ. Math. IHÉS, 90, 1999, 45–143 | DOI | MR | Zbl

[16] H. Matsumura, Commutative Ring Theory, translated from the Japanese by M. Reid., Cambridge Studies in Advanced Mathematics, 8, 2nd ed., Cambridge University Press, Cambridge, 1989 | MR | Zbl

[17] Y. Nisnevich, “Rationally trivial principal homogeneous spaces and arithmetic of reductive group schemes over Dedekind rings”, C. R. Acad. Sci. Paris, Série I, 299:1 (1984), 5–8 | MR | Zbl

[18] Ning Guo, The Grothendieck–Serre conjecture over semi-local Dedekind rings, arXiv: 1902.02315v2

[19] I. Panin, “On Grothendieck–Serre conjecture concerning principal bundles” (Rio de Janeiro, 2018), Proc. Intern. Congress of Math., 201–222 | MR

[20] I. Panin, “Nice triples and moving lemmas for motivic spaces”, Izv. Math., 83:4 (2019), 796–829, arXiv: 1707.01755 | DOI | MR | Zbl

[21] I. Panin, “Nice triples and Grothendieck–Serre's conjecture concerning principal $G$-bundles over reductive group schemes”, Duke Math. J., 168:2 (2019), arXiv: 1707.01756 | DOI | MR | Zbl

[22] I. Panin, Proof of Grothendieck–Serre conjecture on principal $G$-bundles over semi-local regular domains containing a finite field, arXiv: 1707.01767

[23] I. Panin, A. Stavrova, N. Vavilov, “On Grothendieck–Serre's conjecture concerning principal $G$-bundles over reductive group schemes: I”, Compositio Math., 151 (2015), 535–567 | DOI | MR | Zbl

[24] J.-P. Serre, “Espaces fibrés algébriques”, Anneaux de Chow et applications, Séminaire Chevalley, 2-e année, Secrétariat mathématique, Paris, 1958 | MR | Zbl

[25] Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4)., Dirigé par M. Artin, A. Grothendieck, J. L. Verdier, avec la collaboration de P. Deligne, B. Saint-Donat, v. 3, Lecture Notes in Mathematics, 305, Théorie des topos et cohomologie étale des schémas, Springer, 1973 | MR

[26] A. Tsybyshev, “A step toward the mixed-characteristic case of the Grothendieck–Serre conjecture”, Algebra i Analiz, 31:1 (2019), 246–254 | MR