Lemma 68.12.3. 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 (65.26.1.1). By Lemma 68.12.2 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. $\square$

Comment #7867 by Anonymous on

It seems to me that the proof implicitly uses some comparison results between modules on small Zariski and small étale sites. I wonder if adding some links to these results might be good (e.g. Lemma 35.8.10 and Lemma 35.10.2 or Lemma 35.10.3).

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