----- > [!proposition] Proposition. ([[equality of subsheaf and sheaf can be tested on stalks]]) > If $\mathcal{F} \subset \mathcal{G}$ is an inclusion of [[sheaf|sheaves]], one has $\mathcal{F}=\mathcal{G} \iff \mathcal{F}_{p}=\mathcal{G}_{p} \ \fa p.$ ^proposition > [!proof]+ Proof. ([[equality of subsheaf and sheaf can be tested on stalks]]) > One direction is immediate. For the other, suppose $\mathcal{F}_{p}=\mathcal{G}_{p}$ for all $p$. WTS $\mathcal{F} \supset \mathcal{G}$. Let $U$ be an open subset of $X$, and let $s \in \mathcal{G}(U)$. By hypothesis, for each $p \in U$ there exists an open [[neighborhood]] $V_{p} \ni p$ such that $s |_{V_{p}} \in \mathcal{F}(V_{p})$. [[covariant functor|Functoriality]] of presheaves guarantees that these restrictions agree on overlaps. Since the $V_{p}$ cover $U$, they glue into a section in $\mathcal{F}(U)$. Since $\mathcal{F} \subset \mathcal{G}$, this is a section in $\mathcal{G}(U)$ as well; locality implies it in fact equals $s$. Thus, $s \in \mathcal{F}(U)$. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```