The Stacks project

Lemma 72.10.13. The property $\mathcal{P}(f) =$“$f$ is flat” is fpqc local on the base.

Proof. We will use Lemma 72.9.4 to prove this. Assumptions (1) and (2) of that lemma follow from Morphisms of Spaces, Lemma 65.30.5. Let $Z' \to Z$ be a surjective flat morphism of affine schemes over $S$. Let $f : X \to Z$ be a morphism of algebraic spaces, and assume that the base change $f' : Z' \times _ Z X \to Z'$ is flat. We have to show that $f$ is flat. Let $U$ be a scheme and let $U \to X$ be surjective and étale. By Morphisms of Spaces, Lemma 65.30.5 again, it is enough to show that $U \to Z$ is flat. Since $f'$ is flat, and since $Z' \times _ Z U$ is a scheme étale over $Z' \times _ Z X$ we conclude (by the same lemma again) that $Z' \times _ Z U \to Z'$ is flat. As $\{ Z' \to Z\} $ is an fpqc covering we conclude that $U \to Z$ is flat by Descent, Lemma 35.20.15 as desired. $\square$


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