Lemma 77.3.2 (0CVE)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 77: Flatness on Algebraic Spaces
Section 77.3: Relatively pure modules
Lemma 77.3.2 (cite)

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