The Stacks project

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

  1. $\mathcal{X} \to \mathcal{Y}$ is representable by algebraic spaces, and

  2. $\mathcal{Y}$ 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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