Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains
Annales de l'Institut Fourier, Volume 62 (2012) no. 6, p. 2131-2143
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 .
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 .
DOI : https://doi.org/10.5802/aif.2745
Classification:  11E04,  11E08,  11D88,  14G99
Keywords: 2-dimensional local ring, local-global principle, quadratic forms, complete local domain
     author = {HU, Yong},
     title = {Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {62},
     number = {6},
     year = {2012},
     pages = {2131-2143},
     doi = {10.5802/aif.2745},
     zbl = {pre06159908},
     mrnumber = {3060754},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2012__62_6_2131_0}
Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains. Annales de l'Institut Fourier, Volume 62 (2012) no. 6, pp. 2131-2143. doi : 10.5802/aif.2745. https://aif.centre-mersenne.org/item/AIF_2012__62_6_2131_0/

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