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Comptes Rendus Mathématique
Volume 353, n° 7
pages 573-577 (juillet 2015)
Doi : 10.1016/j.crma.2015.03.017
Received : 12 February 2015 ;  accepted : 30 Mars 2015
Dualité sur un corps local de caractéristique positive à corps résiduel algébriquement clos
Duality over a local field of positive characteristic with algebraically closed residue field
 

Cédric Pépin 1
 Département de mathématiques, Institut Galilée, Université Paris-13, 99, avenue Jean-Baptiste-Clément, 93430 Villetaneuse, France 

Résumé

Soit K un corps discrètement valué complet à corps résiduel k de caractéristique  . On dispose d'une théorie de dualité pour la cohomologie des K -schémas en groupes commutatifs finis dans les cas suivants : K est de caractéristique 0 et k fini (J. Tate, Duality theorems in Galois cohomology over number fields, in : Proceedings ICM 1962), K est de caractéristique p et k fini (S.S. Shatz, Cohomology of Artinian group schemes over local fields, Ann. of Math. (2) 79 (3) (1964) 411–449), K est de caractéristique 0 et k algébriquement clos (L. Bégueri, Dualité sur un corps local à corps résiduel algébriquement clos, Mém. Soc. Math. Fr. 108 (4) (1980)). On présente ici le cas où K est de caractéristique p et k algébriquement clos ; il s'agit d'un résumé du texte détaillé (C. Pépin, Dualité sur un corps local de caractéristique positive à corps résiduel algébriquement clos, prépublication, arXiv:1411.0742v1). Une approche indépendante a été donnée récemment par Suzuki (Duality for local fields and sheaves on the category of fields, prépublication, arXiv:1310.4941v2, 2.7.6 (1) (a)).

The full text of this article is available in PDF format.
Abstract

Let K be a complete discretely valued field with residue field k of characteristic  . There exists a duality theory for the cohomology of finite commutative K -group schemes in the following cases: K has characteristic 0 and k is finite (J. Tate, Duality theorems in Galois cohomology over number fields, in: Proceedings ICM 1962), K has characteristic p and k is finite (S.S. Shatz, Cohomology of Artinian group schemes over local fields, Ann. of Math. (2) 79 (3) (1964) 411–449), K has characteristic 0 and k is algebraically closed (L. Bégueri, Dualité sur un corps local à corps résiduel algébriquement clos, Mém. Soc. Math. Fr. 108 (4) (1980)). Here we present the case where K has characteristic p and k is algebraically closed; this is a summary of the detailed text (C. Pépin, Dualité sur un corps local de caractéristique positive à corps résiduel algébriquement clos, prepublication, arXiv:1411.0742v1). An independent approach has been given recently by Suzuki (Duality for local fields and sheaves on the category of fields, prepublication, arXiv:1310.4941v2, 2.7.6 (1) (a)).

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1  Avec le soutien du projet ANR PerCoLaTor ANR-14-CE25.


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