The Stacks project

Proof. By Lemma 94.4.3 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 94.14.2 we know that

is an algebraic stack. It follows from Lemma 94.15.1 and Algebraic Stacks, Lemma 91.15.5 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 94.12.7 and 94.15.3 and the surjectivity a consequence of Lemma 94.15.4. 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 91.10.9 and More on Morphisms of Spaces, Lemma 73.19.6). Applying Lemmas 94.5.3, 94.6.3, and 94.7.3 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

1. limit preserving on objects: this is Lemma 94.12.7 for $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}$ and we combine it with Lemmas 94.15.1, 94.5.4, and 94.5.2 to get it for $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}$,

2. formally smooth on objects by Lemma 94.15.3, and

3. surjective on objects by Lemma 94.15.4.

Using Lemmas 94.5.2, 94.6.2, and 94.7.2 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 94.5.3, 94.6.3, and 94.7.3 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 91.10.9) the infinitesimal lifting criterion (More on Morphisms of Spaces, Lemma 73.19.6) we see that $\mathcal{V} \to \mathcal{Y}$ is smooth and we win. $\square$