The following theorem is one of the main results of this chapter.

**Proof.**
By Lemma 97.4.3 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 97.14.2 we know that

\[ \coprod \nolimits _{d \geq 1} \mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) \]

is an algebraic stack. It follows from Lemma 97.15.1 and Algebraic Stacks, Lemma 94.15.5 that

\[ \coprod \nolimits _{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \]

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

\[ \mathcal{V} \longrightarrow \coprod \nolimits _{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}). \]

We claim that the composition

\[ \mathcal{V} \longrightarrow \coprod \nolimits _{d \geq 1} \mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \longrightarrow \mathcal{Y} \]

is smooth and surjective which finishes the proof of the theorem. In fact, the smoothness will be a consequence of Lemmas 97.12.7 and 97.15.3 and the surjectivity a consequence of Lemma 97.15.4. 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 94.10.9 and More on Morphisms of Spaces, Lemma 76.19.6). Applying Lemmas 97.5.3, 97.6.3, and 97.7.3 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 97.12.7 for $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}$ and we combine it with Lemmas 97.15.1, 97.5.4, and 97.5.2 to get it for $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}$,

formally smooth on objects by Lemma 97.15.3, and

surjective on objects by Lemma 97.15.4.

Using Lemmas 97.5.2, 97.6.2, and 97.7.2 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 97.5.3, 97.6.3, and 97.7.3 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 94.10.9) the infinitesimal lifting criterion (More on Morphisms of Spaces, Lemma 76.19.6) we see that $\mathcal{V} \to \mathcal{Y}$ is smooth and we win.
$\square$

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