---- $(X, \mathcal{O}_{X})$ is a [[ringed space]]. > [!definition] Definition. ([[subsheaf of modules generated by local sections]]) > Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_{X}$-[[sheaf of modules|modules]]. Let $f_{i} \in \Gamma(U_{i}, \mathcal{F})$, $i \in I$, be a collection of local sections of $\mathcal{F}$ supported on $\{ U_{i} \}_{i \in I}$. We define the **subsheaf generated by $\{ f_{i} \}_{i \in I}$** to be the [[sheaf]] [[sheafification|associated to]] the [[presheaf|subpresheaf]] $U \mapsto \left\{ \text{finite sums } \sum_{i} s_{i} \cdot (f_{i} |_{U}) \ : \ U \subset U_{i}, s_{i} \in \mathcal{O}_{X}(U) \right\}.$ This is the unique smallest [[subsheaf]] of $\mathcal{O}_{X}$-[[sheaf of modules|modules]] of $\mathcal{F}$ containing for which each $s_{i}$ corresponds to a local section. ^definition (stacks project lemma 17.4.4) ---- #### ---- #### 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 > ```