The following theorem is one of the main results of this chapter.
Theorem 96.16.1. 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 representable by an algebraic space, and
$F$ is representable by algebraic spaces, surjective, flat and locally of finite presentation,
then $\mathcal{Y}$ is an algebraic stack.
Proof. By Lemma 96.4.3 we see that the diagonal of $\mathcal{Y}$ is representable by algebraic spaces. Hence we only need to verify the existence of a $1$-morphism $f : \mathcal{V} \to \mathcal{Y}$ of stacks in groupoids over $(\mathit{Sch}/S)_{fppf}$ with $\mathcal{V}$ representable and $f$ surjective and smooth. By Lemma 96.14.2 we know that
is an algebraic stack. It follows from Lemma 96.15.1 and Algebraic Stacks, Lemma 93.15.5 that
is an algebraic stack as well. Choose a representable stack in groupoids $\mathcal{V}$ over $(\mathit{Sch}/S)_{fppf}$ and a surjective and smooth $1$-morphism
We claim that the composition
is smooth and surjective which finishes the proof of the theorem. In fact, the smoothness will be a consequence of Lemmas 96.12.7 and 96.15.3 and the surjectivity a consequence of Lemma 96.15.4. We spell out the details in the following paragraph.
By construction $\mathcal{V} \to \coprod \nolimits _{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ is representable by algebraic spaces, surjective, and smooth (and hence also locally of finite presentation and formally smooth by the general principle Algebraic Stacks, Lemma 93.10.9 and More on Morphisms of Spaces, Lemma 75.19.6). Applying Lemmas 96.5.3, 96.6.3, and 96.7.3 we see that $\mathcal{V} \to \coprod \nolimits _{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y})$ is limit preserving on objects, formally smooth on objects, and surjective on objects. The $1$-morphism $\coprod \nolimits _{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}$ is
limit preserving on objects: this is Lemma 96.12.7 for $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}$ and we combine it with Lemmas 96.15.1, 96.5.4, and 96.5.2 to get it for $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}$,
formally smooth on objects by Lemma 96.15.3, and
surjective on objects by Lemma 96.15.4.
Using Lemmas 96.5.2, 96.6.2, and 96.7.2 we conclude that the composition $\mathcal{V} \to \mathcal{Y}$ is limit preserving on objects, formally smooth on objects, and surjective on objects. Using Lemmas 96.5.3, 96.6.3, and 96.7.3 we see that $\mathcal{V} \to \mathcal{Y}$ is locally of finite presentation, formally smooth, and surjective. Finally, using (via the general principle Algebraic Stacks, Lemma 93.10.9) the infinitesimal lifting criterion (More on Morphisms of Spaces, Lemma 75.19.6) we see that $\mathcal{V} \to \mathcal{Y}$ is smooth and we win. $\square$
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