Lemma 37.61.3. Let $X \to Y$ be a morphism of schemes such that $X \to X \times _ Y X$ is flat. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. If $\mathcal{F}$ is flat over $Y$, then $\mathcal{F}$ is flat over $X$.

**Proof.**
Let $x \in X$ with image $y = f(x)$ in $Y$. Since $X \to X \times _ Y X$ is flat, we see that $\mathcal{O}_{X, x} \otimes _{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x} \to \mathcal{O}_{X, x}$ is flat. Hence the result follows from More on Algebra, Lemma 15.104.2 and the definitions.
$\square$

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