The Stacks project

Lemma 69.12.2. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. The following are equivalent

1. $\mathcal{F}$ is coherent,

2. $\mathcal{F}$ is a quasi-coherent, finite type $\mathcal{O}_ X$-module,

3. $\mathcal{F}$ is a finitely presented $\mathcal{O}_ X$-module,

4. for any étale morphism $\varphi : U \to X$ where $U$ is a scheme the pullback $\varphi ^*\mathcal{F}$ is a coherent module on $U$, and

5. there exists a surjective étale morphism $\varphi : U \to X$ where $U$ is a scheme such that the pullback $\varphi ^*\mathcal{F}$ is a coherent module on $U$.

In particular $\mathcal{O}_ X$ is coherent, any invertible $\mathcal{O}_ X$-module is coherent, and more generally any finite locally free $\mathcal{O}_ X$-module is coherent.