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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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