The Stacks project

Lemma 29.25.13. Let $h : X \to Y$ be a morphism of schemes over $S$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $Y$. Let $x \in X$ with $y = h(x) \in Y$. If $h$ is flat at $x$, then

\[ \mathcal{G}\text{ flat over }S\text{ at }y \Leftrightarrow h^*\mathcal{G}\text{ flat over }S\text{ at }x. \]

In particular: If $h$ is surjective and flat, then $\mathcal{G}$ is flat over $S$, if and only if $h^*\mathcal{G}$ is flat over $S$. If $h$ is surjective and flat, and $X$ is flat over $S$, then $Y$ is flat over $S$.

Proof. You can prove this by applying Algebra, Lemma 10.39.9. Here is a direct proof. Let $s \in S$ be the image of $y$. Consider the local ring maps $\mathcal{O}_{S, s} \to \mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$. By assumption the ring map $\mathcal{O}_{Y, y} \to \mathcal{O}_{X, x}$ is faithfully flat, see Algebra, Lemma 10.39.17. Let $N = \mathcal{G}_ y$. Note that $h^*\mathcal{G}_ x = N \otimes _{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x}$, see Sheaves, Lemma 6.26.4. Let $M' \to M$ be an injection of $\mathcal{O}_{S, s}$-modules. By the faithful flatness mentioned above we have

\begin{align*} \mathop{\mathrm{Ker}}( M' \otimes _{\mathcal{O}_{S, s}} N \to M \otimes _{\mathcal{O}_{S, s}} N) \otimes _{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x} \\ = \mathop{\mathrm{Ker}}( M' \otimes _{\mathcal{O}_{S, s}} N \otimes _{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x} \to M \otimes _{\mathcal{O}_{S, s}} N \otimes _{\mathcal{O}_{Y, y}} \mathcal{O}_{X, x}) \end{align*}

Hence the equivalence of the lemma follows from the second characterization of flatness in Algebra, Lemma 10.39.5. $\square$

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