Lemma 69.12.2 (07UB)—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

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

Proof. To be sure, if $X$ is a locally Noetherian algebraic space and $U \to X$ is an étale morphism, then $U$ is locally Noetherian, see Properties of Spaces, Section 66.7. The lemma then follows from the points (1) – (5) made in Properties of Spaces, Section 66.30 and the corresponding result for coherent modules on locally Noetherian schemes, see Cohomology of Schemes, Lemma 30.9.1. $\square$

The Stacks project

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