Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains
نویسنده
چکیده
— Let R be a 2-dimensional normal excellent henselian local domain in which 2 is invertible and let L and k be its fraction field and residue field respectively. Let ΩR be the set of rank 1 discrete valuations of L corresponding to codimension 1 points of regular proper models of Spec R. We prove that a quadratic form q over L satisfies the local-global principle with respect to ΩR in the following two cases: (1) q has rank 3 or 4; (2) q has rank > 5 and R = A[[y]], where A is a complete discrete valuation ring with a not too restrictive condition on the residue field k, which is satisfied when k is C1. Résumé. — Soit R un anneau local intègre de dimension 2, normal, excellent et hensélien dans lequel 2 est inversible. Soient L son corps de fractions et k son corps résiduel. Soit ΩR l’ensemble des valuations discrètes de rang 1 de L correspondant aux points de codimension 1 des modèles propres réguliers de Spec R. On démontre qu’une forme quadratique q sur L satisfait le principe local-global par rapport à ΩR dans les deux cas suivants : (1) q est de rang 3 ou 4 ; (2) q est de rang > 5 et R = A[[y]], où A est un anneau de valuation discrète complet, avec une condition sur le corps résiduel k qui est satisfaite lorsque k est C1. 1. Statements of results Let R be a 2-dimensional excellent henselian local domain and let L and k be respectively its fraction field and residue field. Assume that the characteristic of k is not 2. Colliot-Thélène, Ojanguren and Parimala [2] proved that any quadratic form of rank at least 5 over L is isotropic when k is separably closed, and that the local-global principle with respect to all discrete valuations (of
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