The Stacks project

Lemma 67.31.3. Let $S$ be a scheme. Let

\[ \xymatrix{ X' \ar[d]_{f'} \ar[r]_{g'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]

be a cartesian diagram of algebraic spaces over $S$. Let $x' \in |X'|$ with image $x \in |X|$. Let $\mathcal{F}$ be a quasi-coherent sheaf on $X$ and denote $\mathcal{F}' = (g')^*\mathcal{F}$.

  1. If $\mathcal{F}$ is flat at $x$ over $Y$ then $\mathcal{F}'$ is flat at $x'$ over $Y'$.

  2. If $g$ is flat at $f'(x')$ and $\mathcal{F}'$ is flat at $x'$ over $Y'$, then $\mathcal{F}$ is flat at $x$ over $Y$.

In particular, if $\mathcal{F}$ is flat over $Y$, then $\mathcal{F}'$ is flat over $Y'$.

Proof. Choose a scheme $V$ and a surjective étale morphism $V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Choose a scheme $V'$ and a surjective étale morphism $V' \to V \times _ Y Y'$. Then $U' = V' \times _ V U$ is a scheme endowed with a surjective étale morphism $U' = V' \times _ V U \to Y' \times _ Y X = X'$. Pick $u' \in U'$ mapping to $x' \in |X'|$. Then we can check flatness of $\mathcal{F}'$ at $x'$ over $Y'$ in terms of flatness of $\mathcal{F}'|_{U'}$ at $u'$ over $V'$. Hence the lemma follows from More on Morphisms, Lemma 37.15.2. $\square$

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