---
tag: KPKND0K7W6
kind: concept
name: Quasi-coherent sheaf
msc: [14F06]
specializes: [S33B2H3MYD]
sources:
  - url: https://en.wikipedia.org/wiki/Coherent_sheaf
---

## Summary

A quasi-coherent sheaf on a scheme is a sheaf of modules that is locally given by a module over the coordinate ring.
On an affine scheme $\operatorname{Spec} A$, quasi-coherent sheaves are the same as $A$-modules.
They form the basic category of linear objects on a scheme, closed under kernels, cokernels, direct sums and tensor products.

## Explanations

### L1 human

An affine scheme is described by a ring $A$, and many linear objects on it are described by $A$-modules.
A quasi-coherent sheaf is what one gets by gluing such modules over the affine pieces of a scheme.
For example, any ideal of $A$ gives a quasi-coherent sheaf on $\operatorname{Spec} A$, and so does any vector space over a field $k$ on the point $\operatorname{Spec} k$.
Most sheaves of modules that arise in algebraic geometry are quasi-coherent.