The Stacks project

Lemma 37.55.3. Let $f : X \to S$ be a morphism of schemes which is locally of finite type. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module.

  1. If $f$ is locally of finite presentation, then $\mathcal{F}$ is of finite presentation relative to $S$ if and only if $\mathcal{F}$ is of finite presentation.

  2. The morphism $f$ is locally of finite presentation if and only if $\mathcal{O}_ X$ is of finite presentation relative to $S$.

Proof. Follows immediately from the definitions, see discussion following More on Algebra, Definition 15.80.2. $\square$

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