---- > [!definition] Definition. ([[sheaf of modules]]) > Let $(X, \mathcal{O}_{X})$ be a [[ringed space]]. A **sheaf of $\mathcal{O}_{X}$-modules** (or simply an **$\mathcal{O}_{X}$-module**) is a [[sheaf]] $\mathcal{F}$ of [[abelian group|abelian groups]] such that $\mathcal{F}(U)$ has $\mathcal{O}_{X}(U)$-[[module]] structure in a manner compatible with restriction, namely, $(s \cdot m) |_{V}=s |_{V} \cdot m |_{V}$ for $s \in \mathcal{O}_{X}(U)$, $m \in \mathcal{F}(U)$, and open $V \subset U$. > > The [[category]] of $\mathcal{O}_{X}$-[[sheaf of modules|modules]] and [[morphism of sheaves of modules|their morphisms]] is denoted $\mathcal{O}_{X} \text{-}\mathsf{Mod}$. ^definition > [!note] Note. > This is not simply a '[[sheaf]] valued in some new [[category]]', because 'the category keeps changing' — e.g. from $\mathcal{O}_{X}(U)\text{-}\mathsf{Mod}$ to $\mathcal{O}_{X}(V)\text{-}\mathsf{Mod}$ upon restriction. Correspondingly, we will also need to formulate upgraded notions of [[sheaf]] [[morphism of (pre)sheaves|morphism]], [[pushforward sheaf|pushforward]], [[pullback sheaf|pullback]], etc. to include requirements of compatibility with $\mathcal{O}_{X}$-[[module]] structure. ^note > [!basicexample] > [[vector field|Vector fields]] on a [[smooth manifold]] form a [[sheaf]] of $C^{\infty}$-modules. ^basic-example ---- #### ---- #### 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 > ```