Definition 110.40.1. Let $X$ be a locally ringed space. An invertible ${\mathcal O}_ X$-module on $X$ is a sheaf of ${\mathcal O}_ X$-modules ${\mathcal L}$ such that every point has an open neighbourhood $U \subset X$ such that ${\mathcal L}|_ U$ is isomorphic to ${\mathcal O}_ U$ as ${\mathcal O}_ U$-module. We say that ${\mathcal L}$ is trivial if it is isomorphic to ${\mathcal O}_ X$ as a ${\mathcal O}_ X$-module.

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