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**}.