# Formal prime ideals of infinite value and their algebraic resolution

• [1] Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
• [2] Department of Mathematics, New York City College of Technology, 300 Jay street, Brooklyn, NY 11201, USA
• Volume: 19, Issue: 3-4, page 635-649
• ISSN: 0240-2963

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## Abstract

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Suppose that $R$ is a local domain essentially of finite type over a field of characteristic $0$, and $\nu$ a valuation of the quotient field of $R$ which dominates $R$. The rank of such a valuation often increases upon extending the valuation to a valuation dominating $\stackrel{^}{R}$, the completion of $R$. When the rank of $\nu$ is $1$, Cutkosky and Ghezzi handle this phenomenon by resolving the prime ideal of infinite value, but give an example showing that when the rank is greater than $1$, there is no natural ideal in $\stackrel{^}{R}$ that leads to this obstruction. We extend their result on the resolution of prime ideals of infinite value to valuations of arbitrary rank.

## How to cite

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Cutkosky, Steven Dale, and ElHitti, Samar. "Formal prime ideals of infinite value and their algebraic resolution." Annales de la faculté des sciences de Toulouse Mathématiques 19.3-4 (2010): 635-649. <http://eudml.org/doc/115870>.

@article{Cutkosky2010,
abstract = {Suppose that $R$ is a local domain essentially of finite type over a field of characteristic $0$, and $\nu$ a valuation of the quotient field of $R$ which dominates $R$. The rank of such a valuation often increases upon extending the valuation to a valuation dominating $\hat\{R\}$, the completion of $R$. When the rank of $\nu$ is $1$, Cutkosky and Ghezzi handle this phenomenon by resolving the prime ideal of infinite value, but give an example showing that when the rank is greater than $1$, there is no natural ideal in $\hat\{R\}$ that leads to this obstruction. We extend their result on the resolution of prime ideals of infinite value to valuations of arbitrary rank.},
affiliation = {Department of Mathematics, University of Missouri, Columbia, MO 65211, USA; Department of Mathematics, New York City College of Technology, 300 Jay street, Brooklyn, NY 11201, USA},
author = {Cutkosky, Steven Dale, ElHitti, Samar},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {formal prime ideals of infinite value; algebraic resolution; valuation rings},
language = {eng},
number = {3-4},
pages = {635-649},
publisher = {Université Paul Sabatier, Toulouse},
title = {Formal prime ideals of infinite value and their algebraic resolution},
url = {http://eudml.org/doc/115870},
volume = {19},
year = {2010},
}

TY - JOUR
AU - Cutkosky, Steven Dale
AU - ElHitti, Samar
TI - Formal prime ideals of infinite value and their algebraic resolution
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2010
PB - Université Paul Sabatier, Toulouse
VL - 19
IS - 3-4
SP - 635
EP - 649
AB - Suppose that $R$ is a local domain essentially of finite type over a field of characteristic $0$, and $\nu$ a valuation of the quotient field of $R$ which dominates $R$. The rank of such a valuation often increases upon extending the valuation to a valuation dominating $\hat{R}$, the completion of $R$. When the rank of $\nu$ is $1$, Cutkosky and Ghezzi handle this phenomenon by resolving the prime ideal of infinite value, but give an example showing that when the rank is greater than $1$, there is no natural ideal in $\hat{R}$ that leads to this obstruction. We extend their result on the resolution of prime ideals of infinite value to valuations of arbitrary rank.
LA - eng
KW - formal prime ideals of infinite value; algebraic resolution; valuation rings
UR - http://eudml.org/doc/115870
ER -

## References

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