Sheaf of modules

Summary

Let $(X, \mathcal{O}_X)$ be a ringed space. A sheaf of modules on it is a sheaf of abelian groups $\mathcal{F}$ in which each $\mathcal{F}(U)$ is a module over $\mathcal{O}_X(U)$, compatibly with restriction. It is the sheaf-theoretic version of a module over a ring. Vector bundles, ideal sheaves, quasi-coherent and coherent sheaves are all sheaves of modules.