  We study the freeness of the group $\mathrm{Inv}(D)$ of invertible ideals of
an integral domain $D$, and the freeness of some related groups of (fractional)
ideals. We study the relation between $\mathrm{Inv}(D)$ and
$\mathrm{Inv}(D_P)$, in particular in the locally finite case, and we analyze
in more detail the case where $D$ is Noetherian (obtaining a characterization
of when $\mathrm{Inv}(D)$ is free for one-dimensional analytically unramified
Noetherian domains) and where $D$ is Pr\"ufer.
