À toute classe dans le groupe de Brauer d’un corps
Let
Mot clés : algèbres simples centrales, indice, exposant, corps de fonctions de deux variables, surfaces complexes, groupe de Brauer, algèbres d'Azumaya, fibrés vectoriels, transformations élémentaires, déformations
Keywords: central simple algebras, index, exponent, function fields in two variables, complex surfaces, Brauer group, Azumaya algebras, vector bundles, elementary transformations, deformation theory
@incollection{SB_2004-2005__47__379_0, author = {Colliot-Th\'el\`ene, Jean-Louis}, title = {Alg\`ebres simples centrales sur les corps de fonctions de deux variables}, booktitle = {S\'eminaire Bourbaki : volume 2004/2005, expos\'es 938-951}, series = {Ast\'erisque}, note = {talk:949}, pages = {379--413}, publisher = {Soci\'et\'e math\'ematique de France}, number = {307}, year = {2006}, mrnumber = {2296425}, zbl = {1123.14012}, language = {fr}, url = {http://www.numdam.org/item/SB_2004-2005__47__379_0/} }
TY - CHAP AU - Colliot-Thélène, Jean-Louis TI - Algèbres simples centrales sur les corps de fonctions de deux variables BT - Séminaire Bourbaki : volume 2004/2005, exposés 938-951 AU - Collectif T3 - Astérisque N1 - talk:949 PY - 2006 SP - 379 EP - 413 IS - 307 PB - Société mathématique de France UR - http://www.numdam.org/item/SB_2004-2005__47__379_0/ LA - fr ID - SB_2004-2005__47__379_0 ER -
%0 Book Section %A Colliot-Thélène, Jean-Louis %T Algèbres simples centrales sur les corps de fonctions de deux variables %B Séminaire Bourbaki : volume 2004/2005, exposés 938-951 %A Collectif %S Astérisque %Z talk:949 %D 2006 %P 379-413 %N 307 %I Société mathématique de France %U http://www.numdam.org/item/SB_2004-2005__47__379_0/ %G fr %F SB_2004-2005__47__379_0
Colliot-Thélène, Jean-Louis. Algèbres simples centrales sur les corps de fonctions de deux variables, dans Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 949, pp. 379-413. http://www.numdam.org/item/SB_2004-2005__47__379_0/
[A] Structure of Algebras, American Mathematical Society Colloquium Publications, vol. 24, American Mathematical Society, New York, 1939. | DOI | JFM | MR | Zbl
-[Ar1] “Algebraic approximation of structures over complete local rings”, Inst. Hautes Études Sci. Publ. Math. (1969), no. 36, p. 23-58. | EuDML | Numdam | MR | Zbl
-[Ar2] -, “Brauer-Severi varieties”, in Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math., vol. 917, Springer, Berlin, 1982, p. 194-210. | MR | Zbl
[Ar3] -, “Two-dimensional orders of finite representation type”, Manuscripta Math. 58 (1987), no. 4, p. 445-471. | EuDML | MR | Zbl
[AG] “Maximal orders”, Trans. Amer. Math. Soc. 97 (1960), p. 1-24. | DOI | MR | Zbl
& -[Bld] Les corps non commutatifs, Presses Universitaires de France, Vendôme, 1972, Collection Sup : Le Mathématicien, No. 9. | MR | Zbl
-[Blt] “Function fields of generalized Brauer-Severi varieties”, Comm. Algebra 19 (1991), no. 1, p. 97-118. | MR | Zbl
-[BLR] “Formal and rigid geometry. IV. The reduced fibre theorem”, Invent. Math. 119 (1995), no. 2, p. 361-398. | EuDML | MR | Zbl
, & -[Bki] Éléments de mathématique. 23. Première partie : Les structures fondamentales de l'analyse. Livre II : Algèbre. Chapitre 8 : Modules et anneaux semi-simples, Actualités Sci. Ind. no. 1261, Hermann, Paris, 1958. | MR | Zbl
-[BHN] “Beweis eines Hauptsatzes in der Theorie der Algebren. ”, J. reine angew. Math. 167 (1932), p. 399-404. | EuDML | JFM
, & -[CT] “Die Brauersche Gruppe ; ihre Verallgemeinerungen und Anwendungen in der arithmetischen Geometrie”, (Vortragsnotizen, Brauer Tagung, Stuttgart, 22.-24. März 2001), tapuscrit.
-[CTG] -, “Exposant et indice d'algèbres simples centrales non ramifiées”, Enseign. Math. (2) 48 (2002), no. 1-2, p. 127-146, avec un appendice d'Ofer Gabber. | MR | Zbl
[CTGP] “Arithmetic of linear algebraic groups over 2-dimensional geometric fields”, Duke Math. J. 121 (2004), no. 2, p. 285-341. | MR | Zbl
, & -[CTOP] “Quadratic forms over fraction fields of two-dimensional Henselian rings and Brauer groups of related schemes”, in Algebra, arithmetic and geometry (Mumbai, 2000), Tata Inst. Fund. Res. Stud. Math., vol. 16, Tata Inst. Fund. Res., Mumbai, 2002, Part II, p. 185-217. | MR | Zbl
, & -[CTS] “Fibrés quadratiques et composantes connexes réelles”, Math. Ann. 244 (1979), no. 2, p. 105-134. | EuDML | MR | Zbl
& -[D] Algebren, Zweite, korrigierte Auflage. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 41, Springer-Verlag, Berlin, 1968. | DOI | MR | Zbl
-[FS] “Division algebras over Henselian surfaces”, in Ring theory 1989 (Ramat Gan and Jerusalem, 1988/1989), Israel Math. Conf. Proc., vol. 1, Weizmann, Jerusalem, 1989, p. 320-336. | MR | Zbl
& -
[G1] “La
[G2] -, “Cohomologie galoisienne des groupes quasi-déployés sur des corps de dimension cohomologique
[GSz] “Central simple algebras and Galois cohomology”, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, 2006. | DOI | MR | Zbl
& -[Gi] Cohomologie non abélienne, Die Grundlehren der mathematischen Wissenschaften, Band 179, Springer-Verlag, Berlin, 1971. | MR | Zbl
-[GHS] “Families of rationally connected varieties”, J. Amer. Math. Soc. 16 (2003), no. 1, p. 57-67. | MR | Zbl
, & -[Gr] “Le groupe de Brauer. I, II, III.”, in Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968. | Zbl
-[EGA] “Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I”, Inst. Hautes Études Sci. Publ. Math. (1961), no. 11. | Numdam | Zbl
& -[HVG] “Zeros and norm groups of quadratic forms over function fields in one variable over a local non-dyadic field”, J. Ramanujan Math. Soc. 13 (1998), no. 2, p. 85-110. | MR | Zbl
& -[HL] The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997. | DOI | MR | Zbl
& -[I] “Grothendieck's existence theorem in formal geometry (with a letter of J-P. Serre)”, in Advanced School in Basic Algebraic Geometry, ICTP Trieste, 2003, soumis à Contemporary Math. | MR
-[dJ1] “The period-index problem for the Brauer group of an algebraic surface”, Duke Math. J. 123 (2004), no. 1, p. 71-94. | MR | Zbl
-[dJ2] -,“A result of Gabber”, tapuscrit.
[dJS1] “Every rationally connected variety over the function field of a curve has a rational point”, Amer. J. Math. 125 (2003), no. 3, p. 567-580. | MR | Zbl
& -[dJS2] -, “Obtaining reduced fibres after normalized base change”, addendum to [dJS1], tapuscrit.
[Kh] “On “horizontal” invariants attached to quadratic forms”, in Algebra and number theory, p. 21-33, Hindustan Book Agency, Delhi, 2005. | MR | Zbl
-[Kt] “A Hasse principle for two-dimensional global fields”, J. reine angew. Math. 366 (1986), p. 142-181. | EuDML | MR | Zbl
-[Kr] “Hodge-theoretic obstruction to the existence of quaternion algebras”, Bull. London Math. Soc. 35 (2003), no. 1, p. 109-116. | MR | Zbl
-[KRTY] “Bicyclic algebras of prime exponent over function fields”, Trans. Amer. Math. Soc. 358 (2006), no. 6, p. 2579-2610. | MR | Zbl
, , & -[Lie] “Twisted sheaves and the period-index problem”, tapuscrit. | Zbl
-[Li] “Introduction to resolution of singularities”, in Algebraic geometry (Proc. Sympos. Pure Math., Vol. 29, Humboldt State Univ., Arcata, Calif., 1974), Amer. Math. Soc., Providence, R.I., 1975, p. 187-230. | MR | Zbl
-
[M1] “Sur le symbole de norme résiduelle de degré
[M2] -, “Algèbres simples centrales et formes quadratiques”, Izv. Akad. Nauk SSSR Ser. Mat. 55 (1991), no. 1, p. 218-224, en russe.
[MS] “
[Mi] Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. | MR | Zbl
-[OP] “Algebras of prime degree on function fields of surfaces”, tapuscrit.
& -
[PS1] “Isotropy of quadratic forms over function fields of
[PS2] -, “On the length of a quadratic form”, in Algebra and Number Theory, p. 147-157, Hindustan Book Agency, Delhi, 2005. | MR | Zbl
[R] Maximal orders, London Mathematical Society Monographs. New Series, vol. 28, The Clarendon Press, Oxford University Press, Oxford, 2003, corrected reprint of the 1975 original. With a foreword by M. J. Taylor. | MR | Zbl
-[Ro] “The Brauer-Hasse-Noether theorem in historical perspective”, Schriften der Math.-Phys. Klasse der Heidelberger Akademie der Wissenschaften, vol. 15, 2004. | MR | Zbl
-
[Sa2] “Cyclic algebras over
[Sa1] -, “Division algebras over
[S1] Corps locaux, Publications de l'Institut de Mathématique de l'Université de Nancago, VIII, Actualités Sci. Indust., No. 1296. Hermann, Paris, 1962. | MR | Zbl
-[S2] -, Cohomologie galoisienne, Lect. Notes in Math., vol. 5, Springer-Verlag, Berlin, 1994, cinquième édition. | MR | Zbl
[S3] -, Cohomologie galoisienne : progrès et problèmes, Séminaire Bourbaki, Exposé 783, Astérisque, vol. 227, 1995, p. 229-257. | Numdam | MR | Zbl
[T] “Algèbres indécomposables d'exposant premier”, Adv. in Math. 65 (1987), no. 3, p. 205-228. | MR | Zbl
-[TW] “Totally ramified valuations on finite-dimensional division algebras”, Trans. Amer. Math. Soc. 302 (1987), no. 1, p. 223-250. | MR | Zbl
& -