Torsion-free Modules over Commutative Domains of Krull Dimension One

TL;DR

This paper investigates conditions under which the class of modules, formed by direct sums of finitely generated torsion-free modules over Krull dimension one domains, is closed under direct summands, revealing connections to local endomorphism rings and normalization properties.

Contribution

It characterizes when the class of such modules is closed under direct summands in local and noetherian domains, and relates this to properties of the normalization and module genus in broader settings.

Findings

Closure under direct summands linked to local endomorphism rings.

Normalization being local implies closure in noetherian cases.

Modules' isomorphism classes are classified by genus in finite character domains.

Abstract

Let $R$ be a domain of Krull dimension one, we study when the class $\mathcal{F}$ of modules over $R$ that are arbitrary direct sums of finitely generated torsion-free modules is closed under direct summands. If $R$ is local, we show that $\mathcal{F}$ is closed under direct summands if and only if any indecomposable, finitely generated, torsion-free module has local endomorphism ring. If, in addition, $R$ is noetherian this is equivalent to say that the normalization of $R$ is a local ring. If $R$ is an $h$-local domain of Krull dimension $1$ and $\mathcal{F}_R$ is closed under direct summands, then the property is inherited by the localizations of $R$ at maximal ideals. Moreover, any localizations of $R$ at a maximal ideal, except maybe one, satisfies that any finitely generated ideal is $2$-generated. The converse is true when the domain $R$ is, in addition, integrally closed, or noetherian semilocal or noetherian with module-finite normalization. Finally, over a commutative domain of finite character and with no restriction on the Krull dimension, we show that the isomorphism classes of countable generated modules in $\mathcal{F}$ are determined by their genus.