The Stacks project

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 $y_1$.

  1. If $\mathcal{F}_{Y_1}$ is pure above $y_1$, then $\mathcal{F}$ is pure above $y$.

  2. 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 ├ętale over $Y_1 \times _ Y T$. Hence $T_1 \to Y_1$ is quasi-finite at $t_1$ because ├ętale 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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