Globalization of some local properties in Krull domains

Abstract:

Let $R$ be a Krull domain. It is shown that a nonzero locally principal ideal is invertible. This is used to show that ${\text {Cl}}(R)/{\text {Pic}}(R)$ is torsion if and only if ${\text {Cl}}({R_M})$ is torsion for each maximal ideal $M$ of $R$. Here ${\text {Cl}}(R)$ and ${\text {Pic}}(R)$ denote the divisor class group and Picard group of $R$, respectively.