Group-theoretic and topological invariants of completely integrally closed Pr\"ufer domains

TL;DR

This paper explores the algebraic and topological invariants of certain classes of Pr"ufer domains, revealing their relationships with ideal factorization, spectrum topology, and extensions to B"ezout domains, with implications for understanding their structure.

Contribution

It establishes the completion relationship between Inv$(R)$ and Div$(R)$, constructs flat extensions to B"ezout domains, and links invariants to spectrum topology in one-dimensional Pr"ufer domains.

Findings

Div$(R)$ is the completion of Inv$(R)$

Existence of flat extensions to B"ezout domains with isomorphic invariants

Spectrum topology determines invariants in certain domains

Abstract

We consider the lattice-ordered groups Inv$(R)$ and Div$(R)$ of invertible and divisorial fractional ideals of a completely integrally closed Pr\"ufer domain. We prove that Div$(R)$ is the completion of the group Inv$(R)$, and we show there is a faithfully flat extension $S$ of $R$ such that $S$ is a completely integrally closed B\'ezout domain with Div$(R) \cong $ Inv$(S)$. Among the class of completely integrally closed Pr\"ufer domains, we focus on the one-dimensional Pr\"ufer domains. This class includes Dedekind domains, the latter being the one-dimensional Pr\"ufer domains whose maximal ideals are finitely generated. However, numerous interesting examples show that the class of one-dimensional Pr\"ufer domains includes domains that differ quite significantly from Dedekind domains by a number of measures, both group-theoretic (involving Inv$(R)$ and Div$(R)$) and topological (involving the maximal spectrum of $R$). We examine these invariants in connection with factorization properties of the ideals of one-dimensional Pr\"ufer domains, putting special emphasis on the class of almost Dedekind domains, those domains for which every localization at a maximal ideal is a rank one discrete valuation domain, as well as the class of SP-domains, those domains for which every proper ideal is a product of radical ideals. For this last class of domains, we show that if in addition the ring has nonzero Jacobson radical, then the lattice-ordered groups Inv$(R)$ and Div$(R)$ are determined entirely by the topology of the maximal spectrum of $R$, and that the Cantor-Bendixson derivatives of the maximal spectrum reflect the distribution of sharp and dull maximal ideals.