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]
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 ${\Omega }_{R}$ 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 ${\Omega }_{R}$ dans les deux cas suivants : (1) $q$ est de rang 3 ou 4 ; (2) $q$ est de rang $\ge 5$ et $R=A\left[\left[y\right]\right]$, 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 ${\Omega }_{R}$ 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 ${\Omega }_{R}$ in the following two cases: (1) $q$ has rank 3 or 4; (2) $q$ has rank $\ge 5$ and $R=A\left[\left[y\right]\right]$, 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 : 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
@article{AIF_2012__62_6_2131_0,
author = {HU, Yong},
title = {Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains},
journal = {Annales de l'Institut Fourier},
pages = {2131--2143},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {62},
number = {6},
year = {2012},
doi = {10.5802/aif.2745},
mrnumber = {3060754},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/aif.2745/}
}
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. http://archive.numdam.org/articles/10.5802/aif.2745/

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