no. 6
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, Yong1
1 Université Paris-Sud, Département de Mathématiques, Bâtiment 425, 91405, Orsay Cedex (France)
Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2131-2143.

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 Ω R l’ensemble des valuations discrètes de rang 1 de L correspondant aux points de codimension 1 des modèles propres réguliers de SpecR. On démontre qu’une forme quadratique q sur L satisfait le principe local-global par rapport à Ω R 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 C 1 .

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 Ω R be the set of rank 1 discrete valuations of L corresponding to codimension 1 points of regular proper models of SpecR. We prove that a quadratic form q over L satisfies the local-global principle with respect to Ω R 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 C 1 .

DOI : 10.5802/aif.2745
Classification : 11E04, 11E08, 11D88, 14G99
Keywords: 2-dimensional local ring, local-global principle, quadratic forms, complete local domain
Mot clés : anneau local de dimension 2, principe local-global, formes quadratiques, anneau local complet

HU, Yong 1

1 Université Paris-Sud, Département de Mathématiques, Bâtiment 425, 91405, Orsay Cedex (France) 
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HU, Yong. Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains. Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2131-2143. doi : 10.5802/aif.2745. https://aif.centre-mersenne.org/articles/10.5802/aif.2745/

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