Adjacent Ideals to Simple Complete Ideals in Two-Dimensional Regular Local Rings

Abstract

Let R be a two-dimensional regular local ring with infinite residue field, and ℘ be a simple complete residually rational ideal of R of order r which determines R _h . Let 𝒯 be the set of quadratic transforms T of R _h with [T: R _h ] = 1, and 𝒮 the set of simple complete ideals of R of order r which are adjacent to ℘ from below. If R _h is free respectively a satellite, then there exist T* ∈ 𝒯 respectively T*, T** ∈ 𝒯 and a bijective map between the set 𝒮 and the set 𝒯∖{T*} respectively 𝒯∖{T*, T**}.

Key Words:

• Ideals adjacent to an ideal
• Simple complete ideals
• Two-dimensional regular local rings

2000 Mathematics Subject Classification:

• 13A15
• 13B20

Notes

Communicated by I. Swanson.

¹These notes shall be available on the first author's homepage http://math-www.uni-paderborn.de/ ∼karlh/Singularities, probably in May, 2010.