---
tag: G596MHC68T
kind: definition
name: Coherent sheaf
msc: [14F06]
defines: 56RXEJ8Z6S
uses: [ND6XEY68P4, KPKND0K7W6, 05QY1VEBYD]
sources:
  - url: https://en.wikipedia.org/wiki/Coherent_sheaf
---

## Definition

Let $X$ be a locally [[Noetherian scheme|ND6XEY68P4]].
A [[quasi-coherent sheaf|KPKND0K7W6]] $\mathcal{F}$ on $X$ is *coherent* if $X$ has a cover by affine open sets $U = \operatorname{Spec} A$ with the following property.
For each such $U$, $\mathcal{F}|_U \cong \widetilde{M}$ for a [[finitely generated|05QY1VEBYD]] $A$-module $M$.

## Remarks

On $\mathbb{A}^1_k = \operatorname{Spec} k[t]$ over a field $k$, the skyscraper sheaf $\widetilde{M}$ with $M = k[t]/(t)$ is coherent, since $M$ is generated by one element.
On $\operatorname{Spec} \Z$, the sheaf $\widetilde{\Q}$ is quasi-coherent but not coherent.
For it to be coherent, $\Q = \widetilde{\Q}(\operatorname{Spec} \Z)$ would have to be a finitely generated $\Z$-module, and it is not.
If $\mathcal{F}$ is coherent on a locally Noetherian scheme, then $\mathcal{F}(U)$ is finitely generated over $A$ for every affine open $U = \operatorname{Spec} A$.
On a general ringed space, Serre's definition asks that $\mathcal{F}$ is locally finitely generated and that every finitely generated subsheaf over an open set is locally finitely presented.
On a locally Noetherian scheme the two definitions agree.