The Stacks project

Theorem 94.16.1. 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 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

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

is an algebraic stack. It follows from Lemma 94.15.1 and Algebraic Stacks, Lemma 91.15.5 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 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. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 06DC. Beware of the difference between the letter 'O' and the digit '0'.