Local monomialization of transcendental extensions

Cutkosky, Steven Dale

doi:10.5802/aif.2132

Local monomialization of transcendental extensions
[Monomialisations locales des extensions transcendantes]

Cutkosky, Steven Dale ¹

¹ University of Missouri, department of mathematics, Columbia, MO 65211 (USA)

Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1517-1586.

Résumé
Abstract

Soient $R \subset S$ deux anneaux locaux réguliers, essentiellement de type fini sur un corps $k$ de caractéristique zéro. Si $V$ est un anneau de valuation du corps des fractions $K$ de $S$ dominant $S$ , nous montrons qu’il existe des suites de transformés monoidaux (éclatements d’idéaux premiers réguliers) $R \to R_{1}$ et $S \to S_{1}$ le long de $V$ tels que $R_{1} \to S_{1}$ est une application monomiale. Il s’ensuit qu’un morphisme de variétés non singulières peut-être rendu monomial le long d’une valuation après éclatement de sous-variétés non singulières.

Suppose that $R \subset S$ are regular local rings which are essentially of finite type over a field $k$ of characteristic zero. If $V$ is a valuation ring of the quotient field $K$ of $S$ which dominates $S$ , then we show that there are sequences of monoidal transforms (blow ups of regular primes) $R \to R_{1}$ and $S \to S_{1}$ along $V$ such that $R_{1} \to S_{1}$ is a monomial mapping. It follows that a morphism of nonsingular varieties can be made to be a monomial mapping along a valuation, after blow ups of nonsingular subvarieties.

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.2132

Classification : 14E, 13A, 13B
Keywords: Monomialization, monoidal transform, valuation ring, Morphism
Mot clés : monomialisation, transformés monoidaux, anneaux de valuation, morphisme

Affiliations des auteurs :

Cutkosky, Steven Dale ¹

¹ University of Missouri, department of mathematics, Columbia, MO 65211 (USA)

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Cutkosky, Steven Dale. Local monomialization of transcendental extensions. Annales de l'Institut Fourier, Tome 55 (2005) no. 5, pp. 1517-1586. doi : 10.5802/aif.2132. https://www.numdam.org/articles/10.5802/aif.2132/

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