Lemma 94.9.7 (0302)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 94: Algebraic Stacks
Section 94.9: Morphisms representable by algebraic spaces
Lemma 94.9.7 (cite)

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Lemma 94.9.7. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$ . Let $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ be categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ . Let $f : \mathcal{X} \to \mathcal{Y}$ be a $1$ -morphism representable by algebraic spaces. Let $g : \mathcal{Z} \to \mathcal{Y}$ be any $1$ -morphism. Consider the fibre product diagram

$\xymatrix{ \mathcal{Z} \times _{g, \mathcal{Y}, f} \mathcal{X} \ar[r]_-{g'} \ar[d]_{f'} & \mathcal{X} \ar[d]^ f \\ \mathcal{Z} \ar[r]^ g & \mathcal{Y} }$

Then the base change $f'$ is a $1$ -morphism representable by algebraic spaces.

Proof. This is formal. $\square$

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