The Stacks project

Proposition 92.13.3. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $\mathcal{X}$ be an algebraic stack over $S$. The following are equivalent

  1. $\mathcal{X}$ is a stack in setoids,

  2. the canonical $1$-morphism $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is an equivalence, and

  3. $\mathcal{X}$ is representable by an algebraic space.

Proof. The equivalence of (1) and (2) follows from Stacks, Lemma 8.7.2. The implication (3) $\Rightarrow $ (1) follows from Lemma 92.13.2. Finally, assume (1). By Stacks, Lemma 8.6.3 there exists an equivalence $j : \mathcal{X} \to \mathcal{S}_ F$ where $F$ is a sheaf on $(\mathit{Sch}/S)_{fppf}$. By Lemma 92.9.5 the fact that $\Delta _\mathcal {X}$ is representable by algebraic spaces, means that $\Delta _ F : F \to F \times F$ is representable by algebraic spaces. Let $U$ be a scheme and let $x : (\mathit{Sch}/U)_{fppf} \to \mathcal{X}$ be a surjective smooth morphism. The composition $j \circ x : (\mathit{Sch}/U)_{fppf} \to \mathcal{S}_ F$ corresponds to a morphism $h_ U \to F$ of sheaves. By Bootstrap, Lemma 78.5.1 this morphism is representable by algebraic spaces. Hence by Lemma 92.10.4 we conclude that $h_ U \to F$ is surjective and smooth. In particular it is surjective, flat and locally of finite presentation (by Lemma 92.10.9 and the fact that a smooth morphism of algebraic spaces is flat and locally of finite presentation, see Morphisms of Spaces, Lemmas 65.37.5 and 65.37.7). Finally, we apply Bootstrap, Theorem 78.10.1 to see that $F$ is an algebraic space. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 92.13: Algebraic stacks and algebraic spaces

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Your email address will not be published.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 04SZ. Beware of the difference between the letter 'O' and the digit '0'.