---- > [!definition] Definition. ([[free sheaf of modules]]) > Let $(X, \mathcal{O}_{X})$ be a [[ringed space]]. A [[sheaf]] $\mathcal{F}$ of $\mathcal{O}_{X}$-[[sheaf of modules|modules]] is said to be **free** if it is [[morphism of sheaves of modules|isomorphic]] to a [[direct sum of sheaves of modules|direct sum]] $\bigoplus_{i \in I} \mathcal{O}_{X}\overbrace{=}^{\text{notat.}}\mathcal{O}_{X}^{\oplus I}$ for $I$ some index set. If $I$ is finite with $|I|=r$, we say $\mathcal{F}$ has **rank** $r$. ^definition > [!note] Remark. > The directly analogous definition is that of a [[free module|free]] [[module]]. ^note ---- #### - [ ] is there a [[universal property]] version of this definition hiding? ---- #### 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 > ```