A proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing infinite fields
Publications Mathématiques de l'IHÉS, Tome 122 (2015), pp. 169-193.

Let R be a regular local ring containing an infinite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial if it is trivial over the fraction field of R. In other words, if K is the fraction field of R, then the map of non-abelian cohomology pointed sets

He´t1(R,G)He´t1(K,G)
induced by the inclusion of R into K has a trivial kernel.

DOI : 10.1007/s10240-015-0075-z
Mots clés : Algebraic Group, Group Scheme, Principal Bundle, Monic Polynomial, Closed Subscheme
Fedorov, Roman 1 ; Panin, Ivan 2

1 Mathematics Department, Kansas State University 138 Cardwell Hall 66506 Manhattan KS USA
2 Steklov Institute of Mathematics at St.-Petersburg Fontanka 27 191023 St.-Petersburg Russia
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Fedorov, Roman; Panin, Ivan. A proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing infinite fields. Publications Mathématiques de l'IHÉS, Tome 122 (2015), pp. 169-193. doi : 10.1007/s10240-015-0075-z. http://archive.numdam.org/articles/10.1007/s10240-015-0075-z/

[Che] V. Chernousov, Variations on a theme of groups splitting by a quadratic extension and Grothendieck–Serre conjecture for group schemes F4 with trivial g3 invariant, Doc. Math. Extra volume: Andrei A. Suslin sixtieth birthday (2010), 147–169.

[CTO] Colliot-Thélène, J.-L.; Ojanguren, M. Espaces principaux homogènes localement triviaux, Publ. Math. IHÉS, Volume 75 (1992), pp. 97-122 | DOI | Numdam | Zbl

[CTS] Colliot-Thélène, J.-L.; Sansuc, J.-J. Principal homogeneous spaces under flasque tori: Applications, J. Algebra, Volume 106 (1987), pp. 148-205 | DOI | MR | Zbl

[DG] Demazure, M.; Grothendieck, A. Schémas en Groupes. III: Structure des Schémas en Groupes Réductifs. Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck (1970)

[Fed] R. Fedorov, Affine Grassmannians of group schemes and exotic principal bundles over affine lines, ArXiv e-prints | arXiv

[Gab] Ofer, G. Affine analog of the proper base change theorem, Isr. J. Math., Volume 87 (1994), pp. 325-335 | DOI | Zbl

[Gil1] Gille, P. Torseurs sur la droite affine, Transform. Groups, Volume 7 (2002), pp. 231-245 | DOI | MR | Zbl

[Gil2] Gille, P. Errata: “Torsors on the affine line” (French) [Transform. Groups, 7 (2002), 231–245], Transform. Groups, Volume 10 (2005), pp. 267-269 | DOI | MR

[Gil3] Gille, P. Le problème de Kneser–Tits, Séminaire Bourbaki, 2007/2008, Exp. No. 983 (2009), pp. 39-81

[Gro1] Grothendieck, A. Torsion homologique et sections rationnelles, Anneaux de Chow et Applications, Exp. No. 5 (1958), pp. 1-29

[Gro2] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude local des schémas et des morphismes des schémas, troisième partie, Publ. Math. IHÉS, Volume 28 (1966), p. 255 | DOI | MR

[Gro3] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas, quatrième partie, Publ. Math. IHÉS, Volume 32 (1967), p. 361 | MR

[Gro4] Grothendieck, A. Le groupe de Brauer. II. Théorie cohomologique, Dix Exposés sur la Cohomologie des Schémas (1968), pp. 67-87

[Gro5] Grothendieck, A. Technique de descente et théorèmes d’existence en géometrie algébrique. I. Généralités. Descente par morphismes fidèlement plats, Séminaire Bourbaki, Vol. 5, Exp. No. 190 (1995), pp. 299-327

[Mat] Matsumura, H. Commutative Ring Theory (1989) (translated from the Japanese by M. Reid) | Zbl

[Mer] Merkur’ev, A. S. The norm principle for algebraic groups, St. Petersburg Math. J., Volume 7 (1996), pp. 243-264 | MR

[MV] Morel, F.; Voevodsky, V. A1-homotopy theory of schemes, Publ. Math. IHÉS, Volume 90 (2001), pp. 45-143 (1999) | DOI

[Nis1] Nisnevich, Y. 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., Volume 299 (1984), pp. 5-8 | MR | Zbl

[Nis2] Nisnevich, Y. 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., Volume 309 (1989), pp. 651-655 | MR | Zbl

[Oja1] Ojanguren, M. Quadratic forms over regular rings, J. Indian Math. Soc., Volume 44 (1982), pp. 109-116 (1980) | MR

[Oja2] Ojanguren, M. Unités représentées par des formes quadratiques ou par des normes réduites, Algebraic K -theory, Part II (1982), pp. 291-299 | DOI

[OP] Ojanguren, M.; Panin, I. Rationally trivial Hermitian spaces are locally trivial, Math. Z., Volume 237 (2001), pp. 181-198 | DOI | MR | Zbl

[OPZ] Ojanguren, M.; Panin, I.; Zainoulline, K. On the norm principle for quadratic forms, J. Ramanujan Math. Soc., Volume 19 (2004), pp. 289-300 | MR | Zbl

[Pan] I. Panin, On Grothendieck–Serre’s conjecture concerning principal G-bundles over reductive group schemes: II. ArXiv e-prints, | arXiv

[Pop] Popescu, D. General Néron desingularization and approximation, Nagoya Math. J., Volume 104 (1986), pp. 85-115 | MR | Zbl

[PPS] I. Panin, V. Petrov and A. Stavrova, Grothendieck–Serre conjecture for adjoint groups of types E_6 and E_7 and for certain classical groups, ArXiv e-prints, | arXiv

[PS1] Panin, I. A.; Suslin, A. A. On a conjecture of Grothendieck concerning Azumaya algebras, St. Petersburg Math. J., Volume 9 (1998), pp. 851-858 | MR

[PS2] V. Petrov and A. Stavrova, Grothendieck–Serre conjecture for groups of type F4 with trivial f3 invariant, ArXiv e-prints, | arXiv

[PSV] Panin, I.; Stavrova, A.; Vavilov, N. On Grothendieck–Serre’s conjecture concerning principal G-bundles over reductive group schemes: I, Compos. Math., Volume 151 (2015), pp. 535-567 | DOI | MR

[Rag1] Raghunathan, M. S. Principal bundles admitting a rational section, Invent. Math., Volume 116 (1994), pp. 409-423 | DOI | MR | Zbl

[Rag2] Raghunathan, M. S. Erratum: “Principal bundles admitting a rational section” [Invent. Math., 116 (1994), 409–423], Invent. Math., Volume 121 (1995), p. 223 | DOI | MR | Zbl

[Ray] Raynaud, M. Anneaux Locaux Henséliens (1970) | Zbl

[Ser] Serre, J.-P. Espaces fibrés algébrique, Anneaux de Chow et Applications (1958), pp. 1-37

[SGA] Artin, M.; Grothendieck, A.; Verdier, J. L. Théorie des Topos et Cohomologie étale des Schémas. Tome 3 (1973) (avec la collaboration de P. Deligne et B. Saint-Donat)

[Spi] Spivakovsky, M. A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms, J. Am. Math. Soc., Volume 12 (1999), pp. 381-444 | DOI | MR | Zbl

[Swa] Swan, R. G. Néron–Popescu desingularization, Algebra and Geometry (1998), pp. 135-192

[Voe] Voevodsky, V. A1-homotopy theory, Proceedings of the International Congress of Mathematicians, Vol. I (1998), pp. 579-604 (electronic)

[Zai] Zainoulline, K. On Grothendieck conjecture about principal homogeneous spaces for some classical algebraic groups, St. Petersburg Math. J., Volume 12 (2001), pp. 117-143 | MR

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