The Stacks project

Theorem 95.16.1. 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

  1. $\mathcal{X}$ is representable by an algebraic space, and

  2. $F$ is representable by algebraic spaces, surjective, flat and locally of finite presentation,

then $\mathcal{Y}$ is an algebraic stack.

Proof. By Lemma 95.4.3 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 95.14.2 we know that

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

is an algebraic stack. It follows from Lemma 95.15.1 and Algebraic Stacks, Lemma 92.15.5 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 95.12.7 and 95.15.3 and the surjectivity a consequence of Lemma 95.15.4. 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 92.10.9 and More on Morphisms of Spaces, Lemma 74.19.6). Applying Lemmas 95.5.3, 95.6.3, and 95.7.3 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 95.12.7 for $\mathcal{H}_ d(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}$ and we combine it with Lemmas 95.15.1, 95.5.4, and 95.5.2 to get it for $\mathcal{H}_{d, lci}(\mathcal{X}/\mathcal{Y}) \to \mathcal{Y}$,

  2. formally smooth on objects by Lemma 95.15.3, and

  3. surjective on objects by Lemma 95.15.4.

Using Lemmas 95.5.2, 95.6.2, and 95.7.2 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 95.5.3, 95.6.3, and 95.7.3 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 92.10.9) the infinitesimal lifting criterion (More on Morphisms of Spaces, Lemma 74.19.6) we see that $\mathcal{V} \to \mathcal{Y}$ is smooth and we win. $\square$


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