Lemma 94.15.4 (05UM)—The Stacks project

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Table of contents
Part 7: Algebraic Stacks
Chapter 94: Algebraic Stacks
Section 94.15: Algebraic stacks, overhauled
Lemma 94.15.4 (cite)

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Lemma 94.15.4. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$ . Let $\mathcal{X} \to \mathcal{Y}$ be a morphism of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$ . Assume that

$\mathcal{X} \to \mathcal{Y}$ is representable by algebraic spaces, and
$\mathcal{Y}$ is an algebraic stack over $S$ .

Then $\mathcal{X}$ is an algebraic stack over $S$ .

Proof. Let $\mathcal{V} \to \mathcal{Y}$ be a surjective smooth $1$ -morphism from a representable stack in groupoids to $\mathcal{Y}$ . This exists by Definition 94.12.1. Then the $2$ -fibre product $\mathcal{U} = \mathcal{V} \times _{\mathcal Y} \mathcal X$ is representable by an algebraic space by Lemma 94.9.8. The $1$ -morphism $\mathcal{U} \to \mathcal X$ is representable by algebraic spaces, smooth, and surjective, see Lemmas 94.9.7 and 94.10.6. By Lemma 94.15.3 we conclude that $\mathcal{X}$ is an algebraic stack. $\square$

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