The Stacks project

Definition 18.32.1. Let $(\mathcal{C}, \mathcal{O})$ be a ringed site.

1. A finite locally free $\mathcal{O}$-module $\mathcal{F}$ is said to have rank $r$ if for every object $U$ of $\mathcal{C}$ there exists a covering $\{ U_ i \to U\} $ of $U$ such that $\mathcal{F}|_{U_ i}$ is isomorphic to $\mathcal{O}_{U_ i}^{\oplus r}$ as an $\mathcal{O}_{U_ i}$-module.

2. An $\mathcal{O}$-module $\mathcal{L}$ is invertible if the functor

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

   is an equivalence.

3. The sheaf $\mathcal{O}^*$ is the subsheaf of $\mathcal{O}$ defined by the rule

   \[ U \longmapsto \mathcal{O}^*(U) = \{ f \in \mathcal{O}(U) \mid \exists g \in \mathcal{O}(U)\text{ such that }fg = 1\} \]

   It is a sheaf of abelian groups with multiplication as the group law.