Lemma 17.4.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. Let $I$ be a set. Let $s_ i \in \Gamma (X, \mathcal{F})$, $i \in I$ be global sections. The sections $s_ i$ generate $\mathcal{F}$ if and only if for all $x\in X$ the elements $s_{i, x} \in \mathcal{F}_ x$ generate the $\mathcal{O}_{X, x}$-module $\mathcal{F}_ x$.

Proof. Omitted. $\square$

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