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Pour R-algèbre S, finie et localement libre, nous proposons une nouvelle définition du foncteur norme , qui en étend l'existence (S peut être ramifiée sur R) et qui en facilite l'emploi (car il est vu comme la solution d'un problème universel).
Mots clefs : corestriction, norme, transfert, restriction de Weil, algèbre d'Azumaya, puissances divisées
A norm functor
Let S be a finite and locally free R-algebra, and let be the usual norm map. We construct a functor which extends both the ``Corestriction'' introduced by C. Riehm when S/R is a finite separable fields extension, and its generalisation by Knus and Ojanguren in case where S is étale over R. Unlike these, the definition we give does not rely upon descent methods, but it rather uses a universal property: N(F) is equipped with a R-polynomial law satisfying the relations , for and , and the couple is universal for these properties. We thus get a well defined functor even if S is ramified over R, but then the image of a projective S-module may fail to be projective over R. Nevertheless, the norm of an invertible S-module is always invertible and for these modules our construction gives the classical one. Moreover, if S is locally of the form R[X] / (P), then N(F) is projective over R for any projective S-module (of finite type); but that fails to be true if is only supposed to be a complete intersection morphism. These points are discussed in some details before we focus on the étale case in order to emphasize the isomorphism between the ``Weil restriction'' and the norm functor (when applied to commutative algebras), and the intricate relations between and for a R-module E.
Class. math. : 13 B 40, 14 E 22, 14 F 20, 55 R 12