Coherent sheaf - Knowlpedia

Definition

Let $(X,\mathcal O_X)$ be a ringed space. An $\mathcal O_X$ -module $\mathcal F$ is coherent if:

$\mathcal F$ is locally of finite type; and
for every open $U\subseteq X$ , every $n\geq0$ , and every morphism $\mathcal O_U^{\oplus n}\to\mathcal F|_U$ , its kernel is locally of finite type.

On a locally Noetherian scheme, a quasi-coherent sheaf is coherent exactly when it is locally of finite type.

Locally free examples

Every finite-rank locally free sheaf is coherent when $\mathcal O_X$ is coherent, in particular on a locally Noetherian scheme. A coherent sheaf need not be locally free; it may be supported on a proper closed subset or have singularities.

References

The Stacks Project Authors, “Coherent modules,” Tag 01BU.