Lemma 94.9.10 (0303)—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 94.9.10 (cite)

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Lemma 94.9.10. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$ . Let $\mathcal{X}_ i, \mathcal{Y}_ i$ be categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$ , $i = 1, 2$ . Let $f_ i : \mathcal{X}_ i \to \mathcal{Y}_ i$ , $i = 1, 2$ be $1$ -morphisms representable by algebraic spaces. Then

$f_1 \times f_2 : \mathcal{X}_1 \times \mathcal{X}_2 \longrightarrow \mathcal{Y}_1 \times \mathcal{Y}_2$

is a $1$ -morphism representable by algebraic spaces.

Proof. Write $f_1 \times f_2$ as the composition $\mathcal{X}_1 \times \mathcal{X}_2 \to \mathcal{Y}_1 \times \mathcal{X}_2 \to \mathcal{Y}_1 \times \mathcal{Y}_2$ . The first arrow is the base change of $f_1$ by the map $\mathcal{Y}_1 \times \mathcal{X}_2 \to \mathcal{Y}_1$ , and the second arrow is the base change of $f_2$ by the map $\mathcal{Y}_1 \times \mathcal{Y}_2 \to \mathcal{Y}_2$ . Hence this lemma is a formal consequence of Lemmas 94.9.9 and 94.9.7. $\square$

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