The Stacks project

Definition 17.25.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. An invertible $\mathcal{O}_ X$-module is a sheaf of $\mathcal{O}_ X$-modules $\mathcal{L}$ such that the functor

\[ \textit{Mod}(\mathcal{O}_ X) \longrightarrow \textit{Mod}(\mathcal{O}_ X),\quad \mathcal{F} \longmapsto \mathcal{L} \otimes _{\mathcal{O}_ X} \mathcal{F} \]

is an equivalence of categories. We say that $\mathcal{L}$ is trivial if it is isomorphic as an $\mathcal{O}_ X$-module to $\mathcal{O}_ X$.

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View Definition 17.25.1 as pdf