The Stacks project

Lemma 97.18.2. Let $S$ be a scheme and let $B$ be an algebraic space over $S$. Let $G$ be a group algebraic space over $B$. Let $X$ be an algebraic space over $B$ and let $a : G \times _ B X \to X$ be an action of $G$ on $X$ over $B$. The quotient stack $[X/G]$ is an algebraic stack if and only if there exists a morphism of algebraic spaces $\varphi : X' \to X$ such that

  1. $G \times _ B X' \to X$, $(g, x') \mapsto a(g, \varphi (x'))$ is a surjection of sheaves, and

  2. the two projections $X'' \to X'$ of the algebraic space $X''$ given by the rule

    \[ T \longmapsto \{ (x'_1, g, x'_2) \in (X' \times _ B G \times _ B X')(T) \mid \varphi (x'_1) = a(g, \varphi (x'_2))\} \]

    are flat and locally of finite presentation.

Proof. This lemma is a special case of Lemma 97.18.1. Namely, the quotient stack $[X/G]$ is by Groupoids in Spaces, Definition 78.20.1 equal to the quotient stack $[X/G \times _ B X]$ of the groupoid in algebraic spaces $(X, G \times _ B X, s, t, c)$ associated to the group action in Groupoids in Spaces, Lemma 78.15.1. There is one small observation that is needed to get condition (1). Namely, the morphism $s : G \times _ B X \to X$ is the second projection and the morphism $t : G \times _ B X \to X$ is the action morphism $a$. Hence the morphism $h : U' \times _{g, U, t} R \to R \xrightarrow {s} U$ from Lemma 97.18.1 corresponds to the morphism

\[ X' \times _{\varphi , X, a} (G \times _ B X) \xrightarrow {\text{pr}_1} X \]

in the current setting. However, because of the symmetry given by the inverse of $G$ this morphism is isomorphic to the morphism

\[ (G \times _ B X) \times _{\text{pr}_1, X, \varphi } X' \xrightarrow {a} X \]

of the statement of the lemma. Details omitted. $\square$

Comments (0)

There are also:

  • 3 comment(s) on Section 97.18: When is a quotient stack algebraic?

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Your email address will not be published.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



View Lemma 97.18.2 as pdf