Groups of invertible ideals of one-dimensional Pr\"ufer domains as groups of integer-valued functions

Abstract

Let $G$ be a one-dimensional $\ell$-subgroup of the group $\mathcal{F}(X,\mathbb{Z})$ of integer-valued functions on a set $X$. We show that $G$ is free under some hypothesis on the spectrum of $G$ and on its quotient groups at the prime ideals. We translate this result in the context of the study of freeness of the group $\mathrm{Inv}(D)$ of invertible ideals of a Pr\"ufer domain $D$: in particular, we introduce the class of \emph{dd-domains} as the class of Pr\"ufer domains having a set $X$ that is dense in $\mathrm{Spec}(D)$ (with respect to the inverse topology) and whose localizations are DVRs. This class is exactly the class of Pr\"ufer domains for which $\mathrm{Inv}(D)$ is isomorphic (as an $\ell$-group) to a subgroup of $\mathcal{F}(X,\mathbb{Z})$.