Algèbres simples centrales sur les corps de fonctions de deux variables  [ Central simple algebras over function fields in two variables ]
Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Talk no. 949, p. 379-413

Let F be a function field in two variables over an algebraically closed field of characteristic zero. The main part of this Bourbaki talk describes A. J. de Jong’s proof (Duke Math. J. 123 (2004) 71-94) that index and exponent coincide for central simple algebras over F. The text follows the simplified approach sketched at the end of de Jong’s paper. Applications to linear algebraic groups are given. A last section surveys results on the comparison between index and exponent for central simple algebras over a function field in one variable over a p-adic field, with application to the isotropy of quadratic forms over such fields.

À toute classe dans le groupe de Brauer d’un corps F sont associés deux entiers, l’indice (degré d’un corps gauche représentant la classe) et l’exposant (ordre de la classe dans le groupe de Brauer). L’exposant divise l’indice, mais ne lui est pas nécessairement égal. Lorsque F est un corps de nombres, c’est un théorème des années 1930 qu’exposant et indice coïncident. A. J. de Jong (Duke Math. J. 123 (2004) 71-94) a montré récemment qu’ils coïncident aussi lorsque F est un corps de fonctions de deux variables sur le corps des complexes. Après des rappels sur le groupe de Brauer (Azumaya et Grothendieck), l’exposé décrit l’essentiel de la démonstration, dans la version simplifiée suggérée à la fin de l’article de de Jong. On donne ensuite quelques conséquences pour les groupes linéaires. Un dernier paragraphe passe en revue des résultats comparant exposant et indice pour les corps de fonctions d’une variable sur les corps p-adiques, avec application à l’isotropie des formes quadratiques sur de tels corps.

Classification:  14F22,  14F05,  14B12,  16K50,  14G99
Keywords: central simple algebras, index, exponent, function fields in two variables, complex surfaces, Brauer group, Azumaya algebras, vector bundles, elementary transformations, deformation theory
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     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},
     author = {Collectif},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {307},
     year = {2006},
     note = {talk:949},
     pages = {379-413},
     zbl = {1123.14012},
     mrnumber = {2296425},
     language = {fr},
     url = {http://www.numdam.org/item/SB_2004-2005__47__379_0}
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Colliot-Thélène, Jean-Louis. Algèbres simples centrales sur les corps de fonctions de deux variables, in Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Talk no. 949, pp. 379-413. http://www.numdam.org/item/SB_2004-2005__47__379_0/

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