Lemma 77.3.4. In Situation 77.2.1. Let (Y_1, y_1) \to (Y, y) be a morphism of pointed algebraic spaces. Assume Y_1 \to Y is flat at y_1.
If \mathcal{F}_{Y_1} is pure above y_1, then \mathcal{F} is pure above y.
If \mathcal{F}_{Y_1} is universally pure above y_1, then \mathcal{F} is universally pure above y.
Proof. This is true because impurities go up along a flat base change, see Lemma 77.2.4. For example part (1) follows because by any impurity (T \to Y, t' \leadsto t, \xi ) of \mathcal{F} above y with T \to Y quasi-finite at t by the lemma leads to an impurity (T_1 \to Y_1, t_1' \leadsto t_1, \xi _1) of the pullback \mathcal{F}_1 of \mathcal{F} to X_1 = Y_1 \times _ Y X over y_1 such that T_1 is étale over Y_1 \times _ Y T. Hence T_1 \to Y_1 is quasi-finite at t_1 because étale morphisms are locally quasi-finite and compositions of locally quasi-finite morphisms are locally quasi-finite (Morphisms of Spaces, Lemmas 67.39.5 and 67.27.3). Similarly for part (2). \square
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