The Stacks project

Lemma 38.16.5. 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. Let $s \in S$. If $\mathcal{O}_{S, s}$ is Noetherian then $\mathcal{F}$ is pure along $X_ s$ if and only if $\mathcal{F}$ is universally pure along $X_ s$.

Proof. First we may replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$, i.e., we may assume that $S$ is Noetherian. Next, use Lemma 38.15.6 and characterization (2) of purity given in discussion following Definition 38.16.1 to conclude. $\square$

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