The Stacks project

Lemma 17.4.6. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_ X$-modules. Given a set $I$, and local sections $s_ i$, $i \in I$ of $\mathcal{F}$. Let $\mathcal{G}$ be the subsheaf generated by the $s_ i$ and let $x\in X$. Then $\mathcal{G}_ x$ is the $\mathcal{O}_{X, x}$-submodule of $\mathcal{F}_ x$ generated by the elements $s_{i, x}$ for those $i$ such that $s_ i$ is defined at $x$.

Proof. This is clear from the construction of $\mathcal{G}$ in the proof of Lemma 17.4.4. $\square$

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