---- > [!definition] Definition. ([[locally free sheaf]]) > Let $(X, \mathcal{O}_{X})$ be a [[ringed space]]. A [[sheaf]] $\mathcal{F}$ of $\mathcal{O}_{X}$-[[sheaf of modules|modules]] is **locally free of rank $r$** if there exists an [[cover|open cover]] $\{ U_{i} \}_{i \in I}$ of $X$ such that $\mathcal{F} |_{U_{i}}$ is [[free sheaf of modules|free]] of rank $r$: $\mathcal{F} |_{U_{i}} \cong \mathcal{O}_{X} ^{\oplus r}|_{U_{i}}.$ > The [[cover]] $\{ U_{i} \}$ is called a **trivializing cover**. > > We say $\mathcal{F}$ is a **line bundle** or **invertible sheaf** if $\mathcal{F}$ is locally free of rank $1$. ^definition > [!basicexample] ^basic-example Let $E \xrightarrow{\pi}B$ be a [[vector bundle]] with typical fiber $V$, $V$ an $\mathbb{R}$-[[vector space]]. With $\mathcal{O}_{B}$ denoting the [[sheaf]] of functions $B \to \mathbb{R}$ on $B$, the [[sheaf of sections of a map|sheaf of]] [[section|sections]] $\mathcal{F}$ of $E \xrightarrow{\pi} B$ is naturally an $\mathcal{O}_{B}$-[[sheaf of modules|module]]. - [ ] finish this discussion from Vakil.. in fact, [[the category of vector bundles over a space is equivalent to that of locally free sheaves on that space]] (not urgent) ---- #### ---- #### 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 > ```