---
tag: S33B2H3MYD
kind: concept
name: Sheaf of modules
msc: [14F06]
sources:
  - url: https://en.wikipedia.org/wiki/Sheaf_of_modules
same_as:
  wikidata: Q18695611
  wikipedia: 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.

## Explanations

### L1 human

On a space, the functions on each open set form a ring.
A sheaf of modules attaches to each open set a module over that ring, in a way that is compatible with shrinking the open set.
The simplest example is the sheaf of functions itself.
Another example is the sheaf of vector fields on a manifold: one can multiply a vector field by a function and get a vector field.