Lemma 67.30.10 (089C)—The Stacks project

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Table of contents
Part 4: Algebraic Spaces
Chapter 67: Morphisms of Algebraic Spaces
Section 67.30: Flat morphisms
Lemma 67.30.10 (cite)

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Lemma 67.30.10. Let $S$ be a scheme. Let $f : Y \to X$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module with scheme theoretic support $Z \subset X$. If $f$ is flat, then $f^{-1}(Z)$ is the scheme theoretic support of $f^*\mathcal{F}$.

Proof. Using the characterization of the scheme theoretic support as given in Lemma 67.15.3 and using the characterization of flat morphisms in terms of étale coverings in Lemma 67.30.5 we reduce to the case of schemes which is Morphisms, Lemma 29.25.14. $\square$

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