Definition 17.4.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. We say that $\mathcal{F}$ is generated by global sections if there exist a set $I$, and global sections $s_ i \in \Gamma (X, \mathcal{F})$, $i \in I$ such that the map

$\bigoplus \nolimits _{i \in I} \mathcal{O}_ X \longrightarrow \mathcal{F}$

which is the map associated to $s_ i$ on the summand corresponding to $i$, is surjective. In this case we say that the sections $s_ i$ generate $\mathcal{F}$.

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