Formes quadratiques et cycles algébriques
Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 941, pp. 113-163.

Introduite par Witt en 1937, la théorie des formes quadratiques sur un corps joue un rôle central dans la démonstration des conjectures de Milnor par Voevodsky via les travaux pionniers de Rost qui y interviennent. Réciproquement, les méthodes de Rost et Voevodsky utilisant la théorie des motifs et les opérations de Steenrod motiviques révolutionnent la théorie des formes quadratiques et ont conduit à la démonstration de résultats de base qui semblaient auparavant inaccessibles. On expliquera notamment comment ces méthodes permettent de démontrer que, si q est une forme quadratique anisotrope dans I n (puissance n-ième de l’idéal d’augmentation de l’anneau de Witt) et que dimq<2 n+1 , alors dimq est de la forme 2 n+1 -2 i pour un entier i{0,,n}.

The theory of quadratic forms over a field was introduced by Witt in 1937. It plays a key rôle in Voevodsky’s proofs of the Milnor conjectures via the pioneering work of Rost. Conversely, the methods of Rost and Voevodsky using the theory of motives and motivic Steenrod operations have had a revolutionary impact on the theory of quadratic forms and have led to proofs of basic results that seemed previously inaccessible. We shall explain, among other things, how these methods yield a proof that, if q is an anisotropic form in I n (the n-th power of the augmentation ideal in the Witt ring) and dimq<2 n+1 , then dimq is of the form 2 n+1 -2 i for some integer i{0,,n}.

Classification : 11E04,  14C25
Mots clés : formes quadratiques, cycles algébriques, motifs
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     author = {Kahn, Bruno},
     title = {Formes quadratiques et cycles alg\'ebriques},
     booktitle = {S\'eminaire Bourbaki : volume 2004/2005, expos\'es 938-951},
     author = {Collectif},
     series = {Ast\'erisque},
     note = {talk:941},
     pages = {113--163},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {307},
     year = {2006},
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Kahn, Bruno. Formes quadratiques et cycles algébriques, dans Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 941, pp. 113-163. http://archive.numdam.org/item/SB_2004-2005__47__113_0/

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