Sous-groupe de Brauer invariant pour un groupe algébrique connexe quelconque
[Invariant Brauer subgroup for an arbitrary connected algebraic group]
Annales de l'Institut Fourier, Volume 74 (2024) no. 2, pp. 627-661.

In this paper, for a smooth variety X equipped with an action of a connected algebraic group G (not necessary linear), we introduce the notion of invariant Brauer sub-group and the notion of invariant étale Brauer–Manin obstruction. Then we prove that this obstruction is equivalent to the étale Brauer–Manin obstruction. This extends the main notion and the key result of author’s previous article and this also extends a result of B. Creutz.

Dans cet article, pour une variété lisse X munie d’une action d’un groupe algébrique connexe G (non nécessairement linéaire), on introduit la notion de sous-groupe de Brauer invariant et la notion d’obstruction de Brauer–Manin étale invariante. Ensuite, on montre que cette obstruction équivaut à l’obstruction de Brauer–Manin étale. Ceci généralise la notion principale et le résultat clé de l’aticle précédent de l’auteur et ceci généralise aussi un résultat de B. Creutz.

Received:
Revised:
Accepted:
Online First:
Published online:
DOI: 10.5802/aif.3590
Classification: 14G12
Mot clés : Groupe algébrique, groupe de Brauer, principe de Hasse.
Keywords: Algebraic group, Brauer group, Hasse principle.
Cao, Yang 1

1 University of Science and Technology of China 96 Jinzhai Road 230026 Hefei (China)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2024__74_2_627_0,
     author = {Cao, Yang},
     title = {Sous-groupe de {Brauer} invariant pour un groupe alg\'ebrique connexe quelconque},
     journal = {Annales de l'Institut Fourier},
     pages = {627--661},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {74},
     number = {2},
     year = {2024},
     doi = {10.5802/aif.3590},
     language = {fr},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3590/}
}
TY  - JOUR
AU  - Cao, Yang
TI  - Sous-groupe de Brauer invariant pour un groupe algébrique connexe quelconque
JO  - Annales de l'Institut Fourier
PY  - 2024
SP  - 627
EP  - 661
VL  - 74
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3590/
DO  - 10.5802/aif.3590
LA  - fr
ID  - AIF_2024__74_2_627_0
ER  - 
%0 Journal Article
%A Cao, Yang
%T Sous-groupe de Brauer invariant pour un groupe algébrique connexe quelconque
%J Annales de l'Institut Fourier
%D 2024
%P 627-661
%V 74
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3590/
%R 10.5802/aif.3590
%G fr
%F AIF_2024__74_2_627_0
Cao, Yang. Sous-groupe de Brauer invariant pour un groupe algébrique connexe quelconque. Annales de l'Institut Fourier, Volume 74 (2024) no. 2, pp. 627-661. doi : 10.5802/aif.3590. https://aif.centre-mersenne.org/articles/10.5802/aif.3590/

[1] Borovoi, Mikhail; Demarche, Cyril Manin obstruction to strong approximation for homogeneous spaces, Comment. Math. Helv., Volume 88 (2013) no. 1, pp. 1-54 | DOI | MR | Zbl

[2] Bröcker, Theodor; tom Dieck, Tammo Representations of compact Lie groups, Graduate Texts in Mathematics, 98, Springer-Verlag, New York, 1995, x+313 pages (Translated from the German manuscript, Corrected reprint of the 1985 translation) | MR | Zbl

[3] Cao, Yang Approximation forte pour les variétés avec une action d’un groupe linéaire, Compos. Math., Volume 154 (2018) no. 4, pp. 773-819 | DOI | MR | Zbl

[4] Cao, Yang Sous-groupe de Brauer invariant et obstruction de descente itérée, Algebra Number Theory, Volume 14 (2020) no. 8, pp. 2151-2183 | DOI | MR | Zbl

[5] Cao, Yang; Demarche, Cyril; Xu, Fei Comparing descent obstruction and Brauer–Manin obstruction for open varieties, Trans. Amer. Math. Soc., Volume 371 (2019) no. 12, pp. 8625-8650 | DOI | MR | Zbl

[6] Cao, Yang; Liang, Yongqi; Xu, Fei Arithmetic purity of strong approximation for homogeneous spaces, J. Math. Pures Appl. (9), Volume 132 (2019), pp. 334-368 | DOI | MR | Zbl

[7] Cao, Yang; Xu, Fei Strong approximation with Brauer–Manin obstruction for groupic varieties, Proc. Lond. Math. Soc. (3), Volume 117 (2018) no. 4, pp. 727-750 | DOI | MR | Zbl

[8] Colliot-Thélène, Jean-Louis; Sansuc, Jean-Jacques La descente sur les variétés rationnelles. II, Duke Math. J., Volume 54 (1987) no. 2, pp. 375-492 | DOI | MR | Zbl

[9] Colliot-Thélène, Jean-Louis; Sansuc, Jean-Jacques Principal homogeneous spaces under flasque tori : applications, J. Algebra, Volume 106 (1987) no. 1, pp. 148-205 | DOI | MR | Zbl

[10] Conrad, Brian Weil and Grothendieck approaches to adelic points, Enseign. Math. (2), Volume 58 (2012) no. 1-2, pp. 61-97 | DOI | MR | Zbl

[11] Creutz, Brendan There are no transcendental Brauer-Manin obstructions on abelian varieties, Int. Math. Res. Not. IMRN (2020) no. 9, pp. 2684-2697 | DOI | MR | Zbl

[12] Demarche, Cyril Méthodes cohomologiques pour l’étude des points rationnels sur les espaces homogènes, Ph. D. Thesis, Université Paris-Sud (2009)

[13] Demarche, Cyril Obstruction de descente et obstruction de Brauer–Manin étale, Algebra Number Theory, Volume 3 (2009) no. 2, pp. 237-254 | DOI | MR | Zbl

[14] Fu, Lei Etale cohomology theory, Nankai Tracts in Mathematics, 14, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015, x+611 pages | DOI | MR | Zbl

[15] Revêtements étales et groupe fondamental (SGA 1) (Grothendieck, A., ed.), Documents Mathématiques (Paris), 3, Société Mathématique de France, Paris, 2003, xviii+327 pages Séminaire de géométrie algébrique du Bois Marie 1960–61. Directed by A. Grothendieck, With two papers by M. Raynaud, Updated and annotated reprint of the 1971 original [Lecture Notes in Math., 224, Springer, Berlin ; MR0354651 (50 #7129)] | MR | Zbl

[16] Harari, David Groupes algébriques et points rationnels, Math. Ann., Volume 322 (2002) no. 4, pp. 811-826 | DOI | MR | Zbl

[17] Harari, David; Skorobogatov, Alexei N. Descent theory for open varieties, Torsors, étale homotopy and applications to rational points (London Math. Soc. Lecture Note Ser.), Volume 405, Cambridge Univ. Press, Cambridge, 2013, pp. 250-279 | DOI | MR | Zbl

[18] Iwasawa, Kenkichi On some types of topological groups, Ann. of Math. (2), Volume 50 (1949), pp. 507-558 | DOI | MR | Zbl

[19] Liu, Qing; Xu, Fei Very strong approximation for certain algebraic varieties, Math. Ann., Volume 363 (2015) no. 3-4, pp. 701-731 | DOI | MR | Zbl

[20] Milne, James S. Étale cohomology, Princeton Mathematical Series, 33, Princeton University Press, Princeton, N.J., 1980, xiii+323 pages | MR | Zbl

[21] Milnor, J. Morse theory, Annals of Mathematics Studies, 51, Princeton University Press, Princeton, N.J., 1963, vi+153 pages (Based on lecture notes by M. Spivak and R. Wells) | DOI | MR | Zbl

[22] Orr, Martin; Skorobogatov, Alexei N.; Valloni, Domenico; Zarhin, Yuri G. Invariant Brauer group of an abelian variety, Israel J. Math., Volume 249 (2022) no. 2, pp. 695-733 | DOI | MR | Zbl

[23] Poonen, Bjorn Insufficiency of the Brauer–Manin obstruction applied to étale covers, Ann. of Math. (2), Volume 171 (2010) no. 3, pp. 2157-2169 | DOI | MR | Zbl

[24] Sansuc, J.-J. Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math., Volume 327 (1981), pp. 12-80 | DOI | MR | Zbl

[25] Skorobogatov, Alexei Torsors and rational points, Cambridge Tracts in Mathematics, 144, Cambridge University Press, Cambridge, 2001, viii+187 pages | DOI | MR | Zbl

[26] Skorobogatov, Alexei Descent obstruction is equivalent to étale Brauer-Manin obstruction, Math. Ann., Volume 344 (2009) no. 3, pp. 501-510 | DOI | MR | Zbl

[27] Skorobogatov, Alexei N. Beyond the Manin obstruction, Invent. Math., Volume 135 (1999) no. 2, pp. 399-424 | DOI | MR | Zbl

[28] Skorobogatov, Alexei N.; Zarhin, Yuri G. The Brauer group and the Brauer–Manin set of products of varieties, J. Eur. Math. Soc. (JEMS), Volume 16 (2014) no. 4, pp. 749-768 | DOI | MR | Zbl

[29] Srinivas, V. Algebraic K-theory, Progress in Mathematics, 90, Birkhäuser Boston, Inc., Boston, MA, 1996, xviii+341 pages | DOI | MR | Zbl

[30] Stoll, Michael Finite descent obstructions and rational points on curves, Algebra Number Theory, Volume 1 (2007) no. 4, pp. 349-391 | DOI | MR | Zbl

[31] Weibel, Charles A. An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, Cambridge, 1994, xiv+450 pages | DOI | MR | Zbl

Cited by Sources: