---
tag: 56RXEJ8Z6S
kind: concept
name: Coherent sheaf
msc: [14F06]
specializes: [KPKND0K7W6]
sources:
  - url: https://en.wikipedia.org/wiki/Coherent_sheaf
same_as:
  wikidata: Q906907
  wikipedia: Coherent sheaf
---

## Summary

A coherent sheaf on a Noetherian scheme is a quasi-coherent sheaf that is locally given by finitely generated modules.
Coherent sheaves are the algebraic analogue of finite-dimensional data: on a proper scheme over a field, their cohomology groups are finite-dimensional.
They include vector bundles, ideal sheaves of closed subschemes, and structure sheaves of closed subschemes.

## Explanations

### L1 human

A coherent sheaf is a quasi-coherent sheaf that is finite in each small region.
On an affine piece it is described by a finitely generated module, so by finitely many generators and relations.
Vector bundles are coherent, but coherent sheaves can also do things vector bundles cannot.
For example, the sheaf that has the value $k$ at the origin of a line and $0$ elsewhere is coherent, with a jump in rank at one point.