The Stacks project

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

\[ U \longmapsto \{ \text{sums } \sum \nolimits _{i \in J} f_ i (s_ i|_ U) \text{ where } J \text{ is finite, } U \subset U_ i \text{ for } i\in J, \text{ and } f_ i \in \mathcal{O}_ X(U) \} \]

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. $\square$

Comments (2)

Comment #5969 by Laurent Moret-Bailly on

Alternatively, can be defined as the image of the canonical map deduced from the 's, where is the inclusion. Not sure this is better but it could be stated as a remark, and in fact this is used to prove Lemma 02UX in case has one element and .

Comment #6148 by on

Yes, I will add this if we ever need it. Thanks!

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