The Stacks project

Lemma 37.58.4. Let $\pi : X \to Y$ be a finite morphism of schemes locally of finite type over a base scheme $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Then $\mathcal{F}$ is of finite presentation relative to $S$ if and only if $\pi _*\mathcal{F}$ is of finite presentation relative to $S$.

Proof. Translation of the result of More on Algebra, Lemma 15.80.3 into the language of schemes. $\square$

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