Lemma 38.17.1. Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module.

If the support of $\mathcal{F}$ is proper over $S$, then $\mathcal{F}$ is universally pure relative to $S$.

If $f$ is proper, then $\mathcal{F}$ is universally pure relative to $S$.

If $f$ is proper, then $X$ is universally pure relative to $S$.

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