Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains

[Principe local-global pour les formes quadratiques sur les corps de fractions d’anneaux henséliens de dimension deux]

HU, Yong

Annales de l'Institut Fourier, Tome 62 (2012), p. 2131-2143 / Harvested from Numdam

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 $\Omega _R$ l’ensemble des valuations discrètes de rang 1 de $L$ correspondant aux points de codimension 1 des modèles propres réguliers de $\operatorname{Spec}R$. On démontre qu’une forme quadratique $q$ sur $L$ satisfait le principe local-global par rapport à $\Omega _R$ dans les deux cas suivants : (1) $q$ est de rang 3 ou 4 ; (2) $q$ est de rang $\ge 5$ 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 $C_1$.

Abstract

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 $\Omega _R$ be the set of rank 1 discrete valuations of $L$ corresponding to codimension 1 points of regular proper models of $\operatorname{Spec}R$. We prove that a quadratic form $q$ over $L$ satisfies the local-global principle with respect to $\Omega _R$ in the following two cases: (1) $q$ has rank 3 or 4; (2) $q$ has rank $\ge 5$ 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 $C_1$.

Détail

Publié le : 2012-01-01

MR 3060754 | Zbl pre06159908

DOI : https://doi.org/10.5802/aif.2745
Classification:  11E04,  11E08,  11D88,  14G99
Mots clés: anneau local de dimension 2, principe local-global, formes quadratiques, anneau local complet

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