The Stacks project

Lemma 38.17.4. Let $f : X \to S$ be a finite type, affine morphism of schemes. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module such that $f_*\mathcal{F}$ is locally projective on $S$, see Properties, Definition 28.21.1. Then $\mathcal{F}$ is universally pure over $S$.

Proof. After reducing to the case where $S$ is the spectrum of a henselian local ring this follows from Lemma 38.14.1. $\square$

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