Globalization of some local properties in Krull domains

Abstract

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