Formes quadratiques et cycles algébriques

Kahn, Bruno

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

Résumé
Abstract

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 $dim q < 2^{n + 1}$ , alors $dim q$ est de la forme $2^{n + 1} - 2^{i}$ pour un entier $i \in {0, \dots, n}$ .

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 $dim q < 2^{n + 1}$ , then $dim q$ is of the form $2^{n + 1} - 2^{i}$ for some integer $i \in {0, \dots, n}$ .

EuDML MR Zbl

Classification : 11E04, 14C25
Mot clés : formes quadratiques, cycles algébriques, motifs
Keywords: quadratic forms, algebraic cycles, motives

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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://www.numdam.org/item/SB_2004-2005__47__113_0/

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