The Stacks project

Lemma 96.8.3. Let $S$ be a scheme. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$. If $\mathcal{X}$ is an algebraic stack and $\Delta : \mathcal{Y} \to \mathcal{Y} \times \mathcal{Y}$ is representable by algebraic spaces, then $F$ is algebraic.

Proof. Choose a representable stack in groupoids $\mathcal{U}$ and a surjective smooth $1$-morphism $\mathcal{U} \to \mathcal{X}$. Let $T$ be a scheme and let $\xi $ be an object of $\mathcal{Y}$ over $T$. The morphism of $2$-fibre products

\[ (\mathit{Sch}/T)_{fppf} \times _{\xi , \mathcal{Y}} \mathcal{U} \longrightarrow (\mathit{Sch}/T)_{fppf} \times _{\xi , \mathcal{Y}} \mathcal{X} \]

is representable by algebraic spaces, surjective, and smooth as a base change of $\mathcal{U} \to \mathcal{X}$, see Algebraic Stacks, Lemmas 93.9.7 and 93.10.6. By our condition on the diagonal of $\mathcal{Y}$ we see that the source of this morphism is representable by an algebraic space, see Algebraic Stacks, Lemma 93.10.11. Hence the target is an algebraic stack by Algebraic Stacks, Lemma 93.15.3. $\square$

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