Finiteness and Skolem Closure of Ideals for Non-Unibranched Domains

Abstract

A one-dimensional, Noetherian, local domain D with maximal ideal 𝔪 and finite residue field was known to be an almost strong Skolem ring if analytically irreducible. It was unknown whether this condition is necessary. We show that it is at least necessary for D to be unibranched. After introducing a general notion of equalizing ideal, we show that, for k large enough, the ideals of the form 𝔐 k, a = {f ∈ Int(D) | f(a) ∈ 𝔪 k }, for a ∈ D, are distinct. This allows to show that the maximal ideals 𝔐 a = {f ∈ Int(D) | f(a) ∈ 𝔪}, although not necessarily distinct, are never finitely generated.