The Stacks project

Lemma 37.55.5. 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. Let $S' \to S$ be a morphism of schemes, set $X' = X \times _ S S'$ and denote $\mathcal{F}'$ the pullback of $\mathcal{F}$ to $X'$. If $\mathcal{F}$ is of finite presentation relative to $S$, then $\mathcal{F}'$ is of finite presentation relative to $S'$.

Proof. Translation of the result of More on Algebra, Lemma 15.80.5 into the language of schemes. $\square$


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