The Stacks project

Lemma 38.18.4. Let $f : X \to S$ be a finite type morphism of schemes. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Assume $\mathcal{F}$ is flat over $S$. In this case $\mathcal{F}$ is pure relative to $S$ if and only if $\mathcal{F}$ is universally pure relative to $S$.

Proof. Immediate consequence of Lemma 38.18.3 and the definitions. $\square$

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