The Stacks project

Lemma 37.58.8. Let $X \to S$ be a morphism of schemes which is locally of finite type. Let $0 \to \mathcal{F}' \to \mathcal{F} \to \mathcal{F}'' \to 0$ be a short exact sequence of quasi-coherent $\mathcal{O}_ X$-modules.

  1. If $\mathcal{F}', \mathcal{F}''$ are finitely presented relative to $S$, then so is $\mathcal{F}$.

  2. If $\mathcal{F}'$ is a finite type $\mathcal{O}_ X$-module and $\mathcal{F}$ is finitely presented relative to $S$, then $\mathcal{F}''$ is finitely presented relative to $S$.

Proof. Translation of the result of More on Algebra, Lemma 15.80.9 into the language of schemes. $\square$

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