Séminaires : Séminaire Général de Logique

Equipe(s) : lm,
Responsables :S. Anscombe, A. Khélif, A. Vignati
Email des responsables : sylvy.anscombe@imj-prg.fr, vignati@imj-prg.fr
Salle : 1013
Adresse :Sophie Germain

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Orateur(s) Max Dickmann - CNRS (Émérite),
Titre Quadratic form theory; from orderable fields to preordered rings
Horaire15:15 à 16:15
RésumeJoint work with F. Miraglia and H. Ribeiro (University of São Paulo, Brazil). This talk will present (in English or French, as the public wishes) a summary of the main results of a joint paper (« Special groups and quadratic forms over rings with non zero-divisor coefficients », 60 pp.), to appear in Fundamenta Math. We develop a theory of quadratic forms with non zero-divisor coefficients over preordered (commutative, unitary) rings , where 2 is invertible and the preorder T satisfies a mild extra condition. In this context we prove that several major results known to hold in classical quadratic form theory over (orderable) fields –e.g., the Arason-Pfister Hauptsatz and Pfister’s localglobal principle-- carry over to any class of preordered rings satisfying a property called NT-quadratic faithfulness. We show that this property holds for many classes of rings frequently met in mathematical practice, such as: — the reduced f-rings and some of their extensions, for which Marshall’s signature conjecture and a vast generalization of Sylvester’s inertia law hold as well; — the reduced, partially ordered Noetherian rings and many of their quotients (a result relevant in real algebraic geometry). The abstract, axiomatic approach to quadratic form theory known as special groups [Dickmann-Miraglia, Memoirs AMS 689 (2000)] is the main tool used in the proofs of these results.
AdresseSophie Germain