Lemma 29.25.4 (0FLM)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 29: Morphisms of Schemes
Section 29.25: Flat morphisms
Lemma 29.25.4 (cite)

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Lemma 29.25.4. Let $f : X \to Y$ be an affine morphism of schemes over a base scheme $S$ . Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$ -module. Then $\mathcal{F}$ is flat over $S$ if and only if $f_*\mathcal{F}$ is flat over $S$ .

Proof. By Lemma 29.25.2 and the fact that $f$ is an affine morphism, this reduces us to the affine case. Say $X \to Y \to S$ corresponds to the ring maps $C \leftarrow B \leftarrow A$ . Let $N$ be the $C$ -module corresponding to $\mathcal{F}$ . Recall that $f_*\mathcal{F}$ corresponds to $N$ viewed as a $B$ -module, see Schemes, Lemma 26.7.3. Thus the result is clear. $\square$

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