Groupe de Brauer non ramifié de quotients par un groupe fini
HTML articles powered by AMS MathViewer
- by J.-L. Colliot-Thélène PDF
- Proc. Amer. Math. Soc. 142 (2014), 1457-1469 Request permission
Abstract:
Soit $k$ un corps, $G$ un groupe fini, $G \hookrightarrow SL_{n,k}$ un plongement. Pour $k$ algébriquement clos, Bogomolov a donné une formule pour le groupe de Brauer non ramifié du quotient $SL_{n,k}/G$. On examine ce que donne sa méthode sur un corps $k$ quelconque (de caractéristique nulle). Par cette méthode purement algébrique, on retrouve et étend des résultats obtenus au moyen de méthodes arithmétiques par Harari et par Demarche, comme la tri- vialité du groupe de Brauer non ramifié pour $k=\mathbf {Q}$ et $G$ d’ordre impair.
[Let $k$ be a field, $G$ a finite group, $G \hookrightarrow SL_{n,k}$ an embedding. For $k$ an algebraically closed field, Bogomolov gave a formula for the unramified Brauer group of the quotient $SL_{n,k}/G$. We develop his method over any characteristic zero field. This purely algebraic method enables us to recover and generalize results of Harari and of Demarche over number fields, such as the triviality of the unramified Brauer group for $k=\mathbf {Q}$ and $G$ of odd order.]
References
- F. A. Bogomolov, The Brauer group of quotient spaces of linear representations, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 3, 485–516, 688 (Russian); English transl., Math. USSR-Izv. 30 (1988), no. 3, 455–485. MR 903621, DOI 10.1070/IM1988v030n03ABEH001024
- J.-L. Colliot-Thélène, Birational invariants, purity and the Gersten conjecture, $K$-theory and algebraic geometry: connections with quadratic forms and division algebras (Santa Barbara, CA, 1992) Proc. Sympos. Pure Math., vol. 58, Amer. Math. Soc., Providence, RI, 1995, pp. 1–64. MR 1327280
- Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, La $R$-équivalence sur les tores, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 2, 175–229 (French). MR 450280
- Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, Principal homogeneous spaces under flasque tori: applications, J. Algebra 106 (1987), no. 1, 148–205. MR 878473, DOI 10.1016/0021-8693(87)90026-3
- Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, La descente sur les variétés rationnelles. II, Duke Math. J. 54 (1987), no. 2, 375–492 (French). MR 899402, DOI 10.1215/S0012-7094-87-05420-2
- Jean-Louis Colliot-Thélène and Jean-Jacques Sansuc, The rationality problem for fields of invariants under linear algebraic groups (with special regards to the Brauer group), Algebraic groups and homogeneous spaces, Tata Inst. Fund. Res. Stud. Math., vol. 19, Tata Inst. Fund. Res., Mumbai, 2007, pp. 113–186. MR 2348904
- Cyril Demarche, Groupe de Brauer non ramifié d’espaces homogènes à stabilisateurs finis, Math. Ann. 346 (2010), no. 4, 949–968 (French, with French summary). MR 2587098, DOI 10.1007/s00208-009-0415-8
- Alexander Grothendieck, Le groupe de Brauer. I. Algèbres d’Azumaya et interprétations diverses, Dix exposés sur la cohomologie des schémas, Adv. Stud. Pure Math., vol. 3, North-Holland, Amsterdam, 1968, pp. 46–66 (French). MR 244269
- David Harari, Quelques propriétés d’approximation reliées à la cohomologie galoisienne d’un groupe algébrique fini, Bull. Soc. Math. France 135 (2007), no. 4, 549–564 (French, with English and French summaries). MR 2439198, DOI 10.24033/bsmf.2545
- Boris Kunyavskiĭ, The Bogomolov multiplier of finite simple groups, Cohomological and geometric approaches to rationality problems, Progr. Math., vol. 282, Birkhäuser Boston, Boston, MA, 2010, pp. 209–217. MR 2605170, DOI 10.1007/978-0-8176-4934-0_{8}
- Nguyen Thi Kim Ngan, Thèse, Université Paris VII, 2010.
- David J. Saltman, Generic Galois extensions and problems in field theory, Adv. in Math. 43 (1982), no. 3, 250–283. MR 648801, DOI 10.1016/0001-8708(82)90036-6
- J.-P. Serre, On the fundamental group of a unirational variety, J. London Math. Soc. 34 (1959), 481–484. MR 109155, DOI 10.1112/jlms/s1-34.4.481
- Jean-Pierre Serre, Cohomologie galoisienne, 5th ed., Lecture Notes in Mathematics, vol. 5, Springer-Verlag, Berlin, 1994 (French). MR 1324577, DOI 10.1007/BFb0108758
- Jean-Pierre Serre, Corps locaux, Publications de l’Université de Nancago, No. VIII, Hermann, Paris, 1968 (French). Deuxième édition. MR 0354618
- Alexei Skorobogatov, Torsors and rational points, Cambridge Tracts in Mathematics, vol. 144, Cambridge University Press, Cambridge, 2001. MR 1845760, DOI 10.1017/CBO9780511549588
Additional Information
- J.-L. Colliot-Thélène
- Affiliation: CNRS, Université Paris-Sud, Département de mathématiques, Bâtiment 425,91405 Orsay Cedex,France
- MR Author ID: 50705
- Email: jlct@math.u-psud.fr
- Received by editor(s): January 25, 2012
- Received by editor(s) in revised form: March 23, 2012, April 17, 2012, May 7, 2012, and May 11, 2012
- Published electronically: February 6, 2014
- Communicated by: Lev Borisov
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 142 (2014), 1457-1469
- MSC (2010): Primary 12G05, 14E08, 14F22, 14M20
- DOI: https://doi.org/10.1090/S0002-9939-2014-11855-7
- MathSciNet review: 3168454