---- > [!definition] Definition. ([[direct sum of sheaves of modules]]) > Let $(X, \mathcal{O}_{X})$ be a [[ringed space]] and let $\{ \mathcal{F}_{i} \}_{i \in I}$ be a family of $\mathcal{O}_{X}$-[[sheaf of modules|modules]]. The [[direct sum of sheaves|direct sum]] $\bigoplus_{i \in I} \mathcal{ F}_{i}$ of the underlying [[sheaf|sheaves]] of [[abelian group|abelian groups]] naturally carries the structure of an $\mathcal{O}_{X}$-[[sheaf of modules|module]]. Explicitly, given $U \subset X$ open, $s \in \mathcal{O}_{X}(U)$ [[module|acts on]] $(m_{i})_{i \in I} \in \bigoplus_{i \in I}\mathcal{F}_{i}$ via $s \cdot (m_{i})_{i \in I}:=(sm_{i})$. ^definition ---- #### ---- #### 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 > ```