Torsion-free Modules over Commutative Domains of Krull Dimension One

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.