Lemma 69.12.3. Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. The category of coherent $\mathcal{O}_ X$-modules is abelian. More precisely, the kernel and cokernel of a map of coherent $\mathcal{O}_ X$-modules are coherent. Any extension of coherent sheaves is coherent.

**Proof.**
Choose a scheme $U$ and a surjective étale morphism $f : U \to X$. Pullback $f^*$ is an exact functor as it equals a restriction functor, see Properties of Spaces, Equation (66.26.1.1). By Lemma 69.12.2 we can check whether an $\mathcal{O}_ X$-module $\mathcal{F}$ is coherent by checking whether $f^*\mathcal{F}$ is coherent. Hence the lemma follows from the case of schemes which is Cohomology of Schemes, Lemma 30.9.2.
$\square$

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