Formal prime ideals of infinite value and their algebraic resolution

Steven Dale Cutkosky; Samar ElHitti

doi:10.5802/afst.1260

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, Série 6, Tome 19 (2010) no. 3-4, pp. 635-649.

Résumé
Abstract

Soit $R$ un domaine local essentiellement de type fini sur un corps de caractéristique $0$ , et $ν$ une valuation du corps de fractions de $R$ qui domine $R$ . Le rang d’une telle valuation augmente souvent en prolongeant la valuation a une valuation dominante $\hat{R}$ qui est la completion de $R$ . Dans le cas où le rang de $ν$ est égal a $1$ , Cutkosky et Ghezzi manipule ce phénomène en résolvant l’ideal premier de valeur infinie. Dans le cas ou le rang est plus grand que 1, ils donnent un example qui montre qu’il n’y a aucun ideal naturel dans $\hat{R}$ qui mene a cette obstruction. Nous généralisons leurs résultats de resolution des ideaux premiers de valeurs infinies aux valuations de rang arbitraire.

Suppose that $R$ is a local domain essentially of finite type over a field of characteristic $0$ , and $ν$ 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 $ν$ 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.

MR Zbl

DOI : 10.5802/afst.1260

Affiliations des auteurs :

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

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Steven Dale Cutkosky; Samar ElHitti. Formal prime ideals of infinite value and their algebraic resolution. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 3-4, pp. 635-649. doi : 10.5802/afst.1260. https://afst.centre-mersenne.org/articles/10.5802/afst.1260/

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F. Cano; M. Fernández-Duque Truncated Local Uniformization of Formal Integrable Differential Forms, Qualitative Theory of Dynamical Systems, Volume 21 (2022) no. 1 | DOI:10.1007/s12346-021-00549-8
Steven Dale Cutkosky Extensions of Valuations to the Henselization and Completion, Acta Mathematica Vietnamica, Volume 44 (2019) no. 1, p. 159 | DOI:10.1007/s40306-018-0267-y
Samar ElHitti Perron Transforms, Communications in Algebra, Volume 42 (2014) no. 5, p. 2003 | DOI:10.1080/00927872.2012.754237

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