The Stacks project

Lemma 76.22.2. Let $S$ be a scheme. Let

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

be a cartesian diagram of algebraic spaces over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_ X$-module. Assume $g$ is flat, $f$ is locally of finite presentation, and $\mathcal{F}$ is locally of finite presentation. Then

\[ \{ x' \in |X'| : (g')^*\mathcal{F}\text{ is flat over }Y'\text{ at }x'\} \]

is the inverse image of the open subset of Theorem 76.22.1 under the continuous map $|g'| : |X'| \to |X|$.

Proof. This follows from Morphisms of Spaces, Lemma 67.31.3. $\square$

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