The Stacks project

Lemma 65.31.5. Let $S$ be a scheme. Let $f : X \to Y$, $g : Y \to Z$ be morphisms of algebraic spaces over $S$. Let $\mathcal{G}$ be a quasi-coherent sheaf on $Y$. Let $x \in |X|$ with image $y \in |Y|$. If $f$ is flat at $x$, then

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

In particular: If $f$ is surjective and flat, then $\mathcal{G}$ is flat over $Z$, if and only if $f^*\mathcal{G}$ is flat over $Z$.

Proof. Pick a geometric point $\overline{x}$ of $X$ and denote $\overline{y}$ the image in $Y$ and $\overline{z}$ the image in $Z$. Via the characterization of flatness in Lemmas 65.31.1 and 65.30.8 and the description of the stalk of $f^*\mathcal{G}$ at $\overline{x}$ of Properties of Spaces, Lemma 64.29.5 the lemma reduces to a purely algebraic question on the local ring maps $\mathcal{O}_{Z, \overline{z}} \to \mathcal{O}_{Y, \overline{y}} \to \mathcal{O}_{X, \overline{x}}$ and the module $\mathcal{G}_{\overline{y}}$. This algebraic statement is Algebra, Lemma 10.39.9. $\square$


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