# The Stacks Project

## Tag 01U6

Lemma 28.24.4. 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 10.38.4. $\square$

The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 4279–4285 (see updates for more information).

\begin{lemma}
\label{lemma-composition-module-flat}
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 $x$.
\end{lemma}

\begin{proof}
See Algebra, Lemma \ref{algebra-lemma-composition-flat}.
\end{proof}

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