Theorem 95.17.2. Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $S$. Assume $s, t$ are flat and locally of finite presentation. Then the quotient stack $[U/R]$ is an algebraic stack over $S$.

Proof. We check the two conditions of Theorem 95.16.1 for the morphism

$(\mathit{Sch}/U)_{fppf} \longrightarrow [U/R].$

The first is trivial (as $U$ is an algebraic space). The second is Lemma 95.17.1. $\square$

## 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Your email address will not be published.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 06FI. Beware of the difference between the letter 'O' and the digit '0'.