Lemma 17.4.4. 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 $s_ i$, $i \in I$ be a collection of local sections of $\mathcal{F}$, i.e., $s_ i \in \mathcal{F}(U_ i)$ for some opens $U_ i \subset X$. There exists a unique smallest subsheaf of $\mathcal{O}_ X$-modules $\mathcal{G}$ such that each $s_ i$ corresponds to a local section of $\mathcal{G}$.

**Proof.**
Consider the subpresheaf of $\mathcal{O}_ X$-modules defined by the rule

Let $\mathcal{G}$ be the sheafification of this subpresheaf. This is a subsheaf of $\mathcal{F}$ by Sheaves, Lemma 6.16.3. Since all the finite sums clearly have to be in $\mathcal{G}$ this is the smallest subsheaf as desired. $\square$

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## Comments (2)

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