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

Yong HU[1]

  • [1] Université Paris-Sud, Département de Mathématiques, Bâtiment 425, 91405, Orsay Cedex (France)

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 6, page 2131-2143
  • ISSN: 0373-0956

Abstract

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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 Spec R . 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 .

How to cite

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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 62.6 (2012): 2131-2143. <http://eudml.org/doc/251135>.

@article{HU2012,
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 $C_1$.},
affiliation = {Université Paris-Sud, Département de Mathématiques, Bâtiment 425, 91405, Orsay Cedex (France)},
author = {HU, Yong},
journal = {Annales de l’institut Fourier},
keywords = {2-dimensional local ring; local-global principle; quadratic forms; complete local domain},
language = {eng},
number = {6},
pages = {2131-2143},
publisher = {Association des Annales de l’institut Fourier},
title = {Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains},
url = {http://eudml.org/doc/251135},
volume = {62},
year = {2012},
}

TY - JOUR
AU - HU, Yong
TI - Local-global principle for quadratic forms over fraction fields of two-dimensional henselian domains
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 6
SP - 2131
EP - 2143
AB - 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 $C_1$.
LA - eng
KW - 2-dimensional local ring; local-global principle; quadratic forms; complete local domain
UR - http://eudml.org/doc/251135
ER -

References

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