Lemma 28.24.4. 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 28.24.2 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 25.7.3. Thus the result is clear. $\square$

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