# The Stacks Project

## Tag 01U8

Lemma 28.24.6. Let $f : X \to S$ be a morphism of schemes. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-modules. Let $g : S' \to S$ be a morphism of schemes. Denote $g' : X' = X_{S'} \to X$ the projection. Let $x' \in X'$ be a point with image $x = g(x') \in X$. If $\mathcal{F}$ is flat over $S$ at $x$, then $(g')^*\mathcal{F}$ is flat over $S'$ at $x'$. In particular, if $\mathcal{F}$ is flat over $S$, then $(g')^*\mathcal{F}$ is flat over $S'$.

Proof. See Algebra, Lemma 10.38.7. $\square$

The code snippet corresponding to this tag is a part of the file morphisms.tex and is located in lines 4303–4314 (see updates for more information).

\begin{lemma}
\label{lemma-base-change-module-flat}
Let $f : X \to S$ be a morphism of schemes.
Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-modules.
Let $g : S' \to S$ be a morphism of schemes.
Denote $g' : X' = X_{S'} \to X$ the projection.
Let $x' \in X'$ be a point with image $x = g(x') \in X$.
If $\mathcal{F}$ is flat over $S$ at $x$, then
$(g')^*\mathcal{F}$ is flat over $S'$ at $x'$.
In particular, if $\mathcal{F}$ is flat over $S$, then
$(g')^*\mathcal{F}$ is flat over $S'$.
\end{lemma}

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

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