# Formal prime ideals of infinite value and their algebraic resolution

Steven Dale Cutkosky^{[1]}; Samar ElHitti^{[2]}

- [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

Annales de la faculté des sciences de Toulouse Mathématiques (2010)

- Volume: 19, Issue: 3-4, page 635-649
- ISSN: 0240-2963

## Access Full Article

top## Abstract

top## How to cite

topCutkosky, 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

top- Abhyankar (S.).— Local uniformization on algebraic surfaces over ground fields of characteristic $p\ne 0$. Annals of Math., 63:491-526 (1956). Zbl0108.16803MR78017
- Abhyankar (S.).— On the valuations centered in a local domain. Amer. J. of Math., 78:321-348 (1956). Zbl0074.26301MR82477
- Abhyankar (S.).— Ramification theoretic methods in algebraic geometry. Bulletin of the Amer. Mathematical Society, 66(4):250-252 (1960). Zbl0101.38201MR1566053
- Abhyankar (S.).— Resolution of singularities of embedded algebraic surfaces. Academic Press (1966). Zbl0147.20504MR217069
- Cossart (V.) and Piltant (O.).— Resolution of singularities of threefolds in positive charateristic I. J. of Algebra, 320:1051-1082 (2008). Zbl1159.14009MR2427629
- Cossart (V.) and Piltant (O.).— Resolution of singularities of threefolds in positive characteristic II. J. of Algebra, 321:1836-1976 (2009). Zbl1173.14012MR2494751
- Cutkosky (S. D.).— Local monomialization and factorization of morphisms. Asterisque, 260 (1999). Zbl0941.14001MR1734239
- Cutkosky (S. D.) and Ghezzi (L.).— Completions of valuation rings. Contemporary Mathematics, 386:13-34 (2005). Zbl1216.13013MR2182768
- ElHitti (S.).— Algebraic resolution of formal ideals along a valuation. PhD thesis, University of Missouri (2008). MR2713680
- Favre (C.) and Jonsson (M.).— The valuative tree. Springer (2004). Zbl1064.14024MR2097722
- Heinzer (W.) and Sally (J.).— Extensions of valuations to the completion of a local domain. J. of pure and applied Algebra, 71:175-185 (1991). Zbl0742.13012MR1117633
- Hironaka (H.).— Resolution of singularities of an algebraic variety over a field of characteristic zero. Annals of Math., 79:109-326 (1964). Zbl0122.38603MR199184
- Knaf (H.) and Kuhlmann (F. V.).— Every place admits local uniformization in a finite extension of the function field. Advances in Math., 221:428-453 (2009). Zbl1221.14016MR2508927
- Kuhlmann (F. V.).— On places of algebraic function fields in arbitrary characteristic. Advances in Math, 188:399-424 (2004). Zbl1134.12304MR2087232
- Hà (H. T.), Ghezzi (L.) and Kashcheyeva (O.).— Toroidalization of generatingsequences in dimension two function fields. J. of Algebra, 301:838-866 (2006). Zbl1170.14003MR2236770
- Matsumura (H.).— Commutative ring theory. Cambridge University Press, Cambridge (1986). Zbl0603.13001MR879273
- Spivakovsky (M.).— Sandwiched singularities and desingularization of surfaces by normalized nash transforms. Ann. of Math, 131:441-491 (1990). Zbl0719.14005MR1053487
- Spivakovsky (M.).— Valuations in function fields of surfaces. Amer. J. Math., 112:107-156 (1990). Zbl0716.13003MR1037606
- Teissier (B.).— Valuations, deformations and toric geometry. In S. Kuhlmann F. V. Kuhlmann and M. Marshall, editors, Valuation theory and its applications II, pages 361-459. AMS and Fields Institute (2003.) Zbl1061.14016MR2018565
- Temkin (M.).— Inseparable local uniformization. Preprint. Zbl1276.14021
- Vaquié (M.).— Extension d’une valuation. Trans. Amer. Math. Soc., 359:3439-3481 (2007). Zbl1121.13006MR2299463
- Zariski (O.).— The reduction of the singularities of an algebraic surface. Annals of Math., 40 (1939). Zbl0021.25303
- Zariski (O.).— Local uniformization of algebraic varieties. Annals of Math., 41:852-896, (1940). Zbl0025.21601MR2864
- Zariski (O.) and Samuel (P.).— Commutative algebra, volume 2. Van Nostrand, Princeton (1960). Zbl0081.26501MR120249

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.