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

In: Algebra, Arithmetic and Geometry with Applications

Author

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  • Silvio Greco
  • Karlheinz Kiyek

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.

Suggested Citation

  • Silvio Greco & Karlheinz Kiyek, 2004. "General Elements of Complete Ideals and Valuations Centered at a Two-dimensional Regular Local Ring," Springer Books, in: Chris Christensen & Avinash Sathaye & Ganesh Sundaram & Chandrajit Bajaj (ed.), Algebra, Arithmetic and Geometry with Applications, pages 381-455, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-18487-1_24
    DOI: 10.1007/978-3-642-18487-1_24
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