The Stacks project

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 ├ętale 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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