The Stacks project

Lemma 29.25.5. Let $X \to Y \to Z$ be morphisms of schemes. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Let $x \in X$ with image $y$ in $Y$. If $\mathcal{F}$ is flat over $Y$ at $x$, and $Y$ is flat over $Z$ at $y$, then $\mathcal{F}$ is flat over $Z$ at $x$.

Proof. See Algebra, Lemma 10.39.4. $\square$

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