Geodesic
Proof of~the Grothendieck--Serre conjecture on principal bundles over regular local rings containing a~field
Izvestiya. Mathematics , Tome 84 (2020) no. 4, pp. 780-795.

Voir la notice de l'article provenant de la source Math-Net.Ru

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 sets induced by the inclusion of $R$ in $K$ has trivial kernel. This result was proved in [1] for regular local rings $R$ containing an infinite field.
Keywords: reductive group schemes, principal bundles, Grothendieck–Serre conjecture. 
@article{IM2_2020_84_4_a6,
     author = {I. A. Panin},
     title = {Proof of~the {Grothendieck--Serre} conjecture on principal bundles over regular local rings containing a~field},
     journal = {Izvestiya. Mathematics },
     pages = {780--795},
     publisher = {mathdoc},
     volume = {84},
     number = {4},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2020_84_4_a6/}
}
TY  - JOUR
AU  - I. A. Panin
TI  - Proof of~the Grothendieck--Serre conjecture on principal bundles over regular local rings containing a~field
JO  - Izvestiya. Mathematics 
PY  - 2020
SP  - 780
EP  - 795
VL  - 84
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2020_84_4_a6/
LA  - en
ID  - IM2_2020_84_4_a6
ER  - 
%0 Journal Article
%A I. A. Panin
%T Proof of~the Grothendieck--Serre conjecture on principal bundles over regular local rings containing a~field
%J Izvestiya. Mathematics 
%D 2020
%P 780-795
%V 84
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2020_84_4_a6/
%G en
%F IM2_2020_84_4_a6
I. A. Panin. Proof of~the Grothendieck--Serre conjecture on principal bundles over regular local rings containing a~field. Izvestiya. Mathematics , Tome 84 (2020) no. 4, pp. 780-795. http://geodesic.mathdoc.fr/item/IM2_2020_84_4_a6/

[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 Études Sci., 122 (2015), 169–193 ; (2013), arXiv: 1211.2678v2 | DOI | MR | Zbl

[2] Schémas en groupes, Séminaire de géométrie algébrique 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. ; v. II, Lecture Notes in Math., 152, ix+654 pp. ; v. III, Lecture Notes in Math., 153, viii+529 pp. | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[3] J. P. Serre, “Les espaces fibrés algébriques”, Anneaux de Chow et applications, Seminaire C. Chevalley; 2e année, Secrétariat mathématique, Paris, 1958, Exp. No. 1, 37 pp. | MR | Zbl

[4] A. Grothendieck, Séminaire Chevalley; 2-e année, Anneaux de Chow et applications, Secrétariat mathématique, Paris, 1958, Exp. No. 5, 29 pp. | MR | Zbl

[5] A. Grothendieck, “Le group de Brauer. II. Théorie cohomologique”, Dix exposés sur la cohomologie de schémas, Adv. Stud. Pure Math., 3, North-Holland, Amsterdam, 1968, 67–87 | MR | Zbl

[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 | MR

[7] I. A. Panin, “Sovershennye troiki i gomotopii otobrazhenii motivnykh prostranstv”, Izv. RAN. Ser. matem., 83:4 (2019), 158–193 ; I. Panin, “Nice triples and moving lemmas for motivic spaces”, Izv. Math., 83:4 (2019), 796–829 ; (2017), arXiv: 1707.01755 | DOI | MR | Zbl | DOI

[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 ; (2017), arXiv: 1707.01756 | DOI | MR | Zbl

[9] I. A. Panin, “Two purity theorems and the Grothendieck-Serre conjecture concerning principal $\mathbf G$-bundles”, Sb. Math., 211:12 (2020), 1777–1794 ; arXiv: 1707.01763 | DOI | DOI | MR

[10] B. Poonen, “Bertini theorems over finite fields”, Ann. of Math. (2), 160:3 (2004), 1099–1127 | DOI | MR | Zbl

[11] F. Charles, B. Poonen, “Bertini irreducibility theorems over finite fields”, J. Amer. Math. Soc., 29:1 (2016), 81–94 ; (2013), arXiv: 1311.4960v1 | DOI | MR | Zbl

[12] M. Ojanguren, I. Panin, “A purity theorem for the Witt group”, Ann. Sci. École Norm. Sup. (4), 32:1 (1999), 71–86 | DOI | MR | Zbl

[13] M. Ojanguren, I. Panin, “Rationally trivial hermitian spaces are locally trivial”, Math. Z., 237:1 (2001), 181–198 | DOI | MR | Zbl

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

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

[16] Ph. Gille, “Le problème de Kneser–Tits”, Séminaire N. Bourbaki, v. 2007/2008, Astérisque, 326, Soc. Math. France, Paris, 2009, Exp. No. 983, vii, 39–81 | MR | Zbl

[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 | DOI | MR | Zbl