The Stacks project

Lemma 18.24.1. Let $(\mathcal{C},\mathcal{O})$ be a ringed site. Let $\mathcal{F}$ be a finitely presented $\mathcal{O}$-module. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be a morphism of $\mathcal{O}$-modules. If $\mathcal{G}$ is finite type, then $\mathop{\mathrm{Coker}}(\varphi )$ is finitely presented.

Proof. Omitted. Hint: See Modules, Lemma 17.11.3. $\square$

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