Definition 17.14.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules.

1. We say $\mathcal{F}$ is locally free if for every point $x \in X$ there exists a set $I$ and an open neighbourhood $x \in U \subset X$ such that $\mathcal{F}|_ U$ is isomorphic to $\bigoplus _{i \in I} \mathcal{O}_ X|_ U$ as an $\mathcal{O}_ X|_ U$-module.

2. We say $\mathcal{F}$ is finite locally free if we may choose the index sets $I$ to be finite.

3. We say $\mathcal{F}$ is finite locally free of rank $r$ if we may choose the index sets $I$ to have cardinality $r$.

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