The Stacks project

Lemma 38.16.4. 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 $s \in S$. Let $(S', s') \to (S, s)$ be a morphism of pointed schemes. If $S' \to S$ is quasi-finite at $s'$ and $\mathcal{F}$ is pure along $X_ s$, then $\mathcal{F}_{S'}$ is pure along $X_{s'}$.

Proof. It $(T \to S', t' \leadsto t, \xi )$ is an impurity of $\mathcal{F}_{S'}$ above $s'$ with $T \to S'$ quasi-finite at $t$, then $(T \to S, t' \to t, \xi )$ is an impurity of $\mathcal{F}$ above $s$ with $T \to S$ quasi-finite at $t$, see Morphisms, Lemma 29.20.12. Hence the lemma follows immediately from the characterization (2) of purity given following Definition 38.16.1. $\square$

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