The Stacks project

Lemma 77.3.2. In Situation 77.2.1.

  1. $\mathcal{F}$ is universally pure above $y$, and

  2. for every morphism $(Y', y') \to (Y, y)$ of pointed algebraic spaces the pullback $\mathcal{F}_{Y'}$ is pure above $y'$.

In particular, $\mathcal{F}$ is universally pure relative to $Y$ if and only if every base change $\mathcal{F}_{Y'}$ of $\mathcal{F}$ is pure relative to $Y'$.

Proof. This is formal. $\square$

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