Lemma 76.6.5. Let $S$ be a scheme. Let $f : X \to Y$ be a decent, finite type morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Assume $\mathcal{F}$ is flat over $Y$. In this case $\mathcal{F}$ is pure relative to $Y$ if and only if $\mathcal{F}$ is universally pure relative to $Y$.

**Proof.**
Immediate consequence of Lemma 76.6.4 and the definitions.
$\square$

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