An extended form of the Grothendieck--Serre conjecture

I. A. Panin

Izvestiya. Mathematics , Tome 86 (2022) no. 4, pp. 782-796.

Résumé

Let $R$ be a regular semi-local integral domain containing a field, $K$ the fraction field of $R$, and $\mu\colon \mathbf{G} \to \mathbf{T}$ an $R$-group scheme morphism between reductive $R$-group schemes which is smooth as a scheme morphism. Suppose that $\mathbf{T}$ is an $R$-torus. Then the map $\mathbf{T}(R)/ \mu(\mathbf{G}(R)) \to \mathbf{T}(K)/ \mu(\mathbf{G}(K))$ is injective and a purity theorem holds. These and other results can be derived from an extended form of the Grothendieck–Serre conjecture proven in the present paper for any such ring $R$.

Détail

Keywords: reductive group schemes, principal bundles, Grothendieck–Serre conjecture, purity theorem.

Bibliographie

[1] Schémas en groupes, Séminaire de géométrie algébrique du Bois Marie 1962/64 (SGA 3), dirigé par M. Demazure, A. Grothendieck, v. III, Lecture Notes in Math., 153, Structure des schémas en groupes réductifs, Rev. reprint, Springer-Verlag, Berlin–New York, 1970 | DOI | MR | Zbl

[2] I. A. Panin, “Two purity theorems and the Grothendieck–Serre conjecture concerning principal $\mathbf G$-bundles”, Mat. Sb., 211:12 (2020), 123–142 ; English transl. Sb. Math., 211:12 (2020), 1777–1794 ; arXiv: 1707.01763 | DOI | MR | Zbl | DOI

[3] I. A. Panin, “Proof of the Grothendieck–Serre conjecture on principal bundles over regular local rings containing a field”, Izv. Ross. Akad. Nauk Ser. Mat., 84:4 (2020), 169–186 ; English transl. Izv. Math., 84:4 (2020), 780–795 | DOI | MR | Zbl | DOI

[4] R. Fedorov and I. Panin, “A proof of Grothendieck–Serre conjecture on principal bundles over regular local rings containing infinite fields”, Publ. Math. Inst. Hautes Études Sci., 122 (2015), 169–193 ; arXiv: 1211.2678v2 | DOI | MR | Zbl

[5] A. Grothendieck, “Technique de descente et théorèmes d'existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats”, Séminaire Bourbaki, Années 1958/59–1959/60. Exposés 169–204, v. 5, Reprint of the 1966 ed., Soc. Math. France, Paris, 1995, Exp. No. 190, 299–327 | MR | Zbl

[6] A. Grothendieck, “Torsion homologique et sections rationnelles”, Séminaire C. Chevalley, 2e année, v. 3, Anneaux de Chow et applications, Secrétariat mathématique, Paris, 1958, Exp. No. 5 | MR | Zbl

[7] J.-P. Serre, “Espaces fibrés algébriques”, Séminaire C. Chevalley, 2e année, v. 3, Anneaux de Chow et applications, Secrétariat mathématique, Paris, 1958, Exp. No. 1 | MR | Zbl

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

[9] J.-L. Colliot-Thélène and J.-J. Sansuc, “Principal homogeneous spaces under flasque tori: applications”, J. Algebra, 106:1 (1987), 148–205 | DOI | MR | Zbl

[10] I. Panin, “Nice triples and the Grothendieck–Serre conjecture concerning principal $\mathrm G$-bundles over reductive group schemes”, Duke Math. J., 168:2 (2019), 351–375 ; arXiv: 1707.01756 | DOI | MR | Zbl

[11] I. Panin, “On Grothendieck–Serre conjecture concerning principal bundles”, Proceedings of the international congress of mathematicians (ICM 2018) (Rio de Janeiro 2018), v. II, Invited lectures, World Sci. Publ., Hackensack, NJ, 2018, 201–221 | DOI | MR | Zbl

[12] Ye. A. Nisnevich, “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., 299:1 (1984), 5–8 | MR | Zbl

[13] I. A. Panin and A. K. Stavrova, “On the Grothendieck–Serre conjecture concerning principal $G$-bundles over semilocal Dedekind domains”, Topics in representation theory of algebras and groups. 29, Zap. Nauchn. Sem. POMI, 443, POMI, St-Petersburg, 2016, 133–146 ; J. Math. Sci. (N.Y.), 222:4 (2017), 453–462 ; arXiv: 1512.00354 | MR | Zbl | DOI

[14] Ning Guo, “The Grothendieck–Serre conjecture over semilocal Dedekind rings”, Transform. Groups, 2020, 1–21, Publ. online ; arXiv: 1902.02315v2 | DOI

[15] V. Chernousov, P. Gille and A. Pianzola, “A classification of torsors over Laurent polynomial rings”, Comment. Math. Helv., 92:1 (2017), 37–55 | DOI | MR | Zbl

[16] I. A. Panin, “On Grothendieck–Serre's conjecture concerning principal $G$-bundles over reductive group schemes: II”, Izv. Ross. Akad. Nauk Ser. Mat., 80:4 (2016), 131–162 ; Izv. Math., 80:4 (2016), 759–790 | DOI | MR | Zbl | DOI

[17] A. Suslin and V. Voevodsky, “Singular homology of abstract algebraic varieties”, Invent. Math., 123:1 (1996), 61–94 | DOI | MR | Zbl

[18] I. Panin, A. Stavrova and 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

[19] I. A. Panin, “Nice triples and moving lemmas for motivic spaces”, Izv. Ross. Akad. Nauk Ser. Mat., 83:4 (2019), 158–193 ; English transl. Izv. Math., 83:4 (2019), 796–829 ; arXiv: 1707.01755 | DOI | MR | Zbl | DOI

[20] D. Popescu, “General Néron desingularization and approximation”, Nagoya Math. J., 104 (1986), 85–115 | DOI | MR | Zbl

[21] R. G. Swan, “Néron–Popescu desingularization”, Algebra and geometry (Taipei 1995), Lect. Algebra Geom., 2, Int. Press, Cambridge, MA, 1998, 135–192 | MR | Zbl