Lemma 17.4.4 (01AP)—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 March 11, 2021 at 03:33

Alternatively, $\mathcal{G}$ can be defined as the image of the canonical map $\bigoplus_{i\in I}j_{i!}\mathcal{O}_{U_i}\to\mathcal{F}$ deduced from the $s_i$ 's, where $j_i:U_i\to X$ 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 $I$ has one element and $\mathcal{O}=\mathbb{Z}$ .

Comment #6148 by Johan on April 30, 2021 at 22:54

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

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