General Elements of Complete Ideals and Valuations Centered at a Two-dimensional Regular Local Ring

Silvio Greco and Karlheinz Kiyek

A chapter in Algebra, Arithmetic and Geometry with Applications, 2004, pp 381-455 from Springer

Abstract: Abstract Let α be a two-dimensional regular local ring with maximal ideal m and residue field κ, and let a be an m-primary complete ideal of α. We give a notion of “general element” of, and we give results on existence and analytic irreducibility of general elements. For example we show that if a is simple then a contains general elements, and any such element is analytically irreducible. We apply these results to the study of the valuation υ p associated to a simple complete -primary ideal p, under the assumption that p is residually rational [e.g. κ algebraically closed]. For this we develop a version of the Hamburger-Noether algorithm which allows to study the quadratic sequences starting from α. Then, among other things, we prove that if f ∈; p is a general element of, then the value semigroup of υ p is equal to the value semigroup of the valuation induced by the integral closure of α/ fα, and we show how to construct a “generating sequence” for υ p.

Keywords: two-dimensional regular local rings; ideal transform; infinitely near points; proximity relations; valuations of the second kind; general elements; intersection multiplicity; complete ideals; simple ideals; value semigroup of a simple ideal; Hamburger-Noether expansion; algebroid curves; symmetric numerical semigroups (search for similar items in EconPapers)
Date: 2004
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-18487-1_24

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DOI: 10.1007/978-3-642-18487-1_24

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