Lemma 67.30.9 (073C)—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 67.30.9 (cite)

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Lemma 67.30.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Then $f$ is flat if and only if the morphism of sites $ (f_{small}, f^\sharp ) : (X_{\acute{e}tale}, \mathcal{O}_ X) \to (Y_{\acute{e}tale}, \mathcal{O}_ Y) $ associated to $f$ is flat.

Proof. Flatness of $(f_{small}, f^\sharp )$ is defined in terms of flatness of $\mathcal{O}_ X$ as a $f_{small}^{-1}\mathcal{O}_ Y$-module. This can be checked at stalks, see Modules on Sites, Lemma 18.39.3 and Properties of Spaces, Theorem 66.19.12. But we've already seen that flatness of $f$ can be checked on stalks, see Lemma 67.30.8. $\square$

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