The Stacks project

Lemma 76.3.3. In Situation 76.2.1. Let $(Y', y') \to (Y, y)$ be a morphism of pointed algebraic spaces. If $Y' \to Y$ is quasi-finite at $y'$ and $\mathcal{F}$ is pure above $y$, then $\mathcal{F}_{Y'}$ is pure above $y'$.

Proof. It $(T \to Y', t' \leadsto t, \xi )$ is an impurity of $\mathcal{F}_{Y'}$ above $y'$ with $T \to Y'$ quasi-finite at $t$, then $(T \to Y, t' \to t, \xi )$ is an impurity of $\mathcal{F}$ above $y$ with $T \to Y$ quasi-finite at $t$, see Morphisms of Spaces, Lemma 66.27.3. Hence the lemma follows immediately from the definition of purity. $\square$


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