The Stacks project

Lemma 78.6.2. Let $B \to S$ be as in Section 78.2. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $G \to U$ be its stabilizer group algebraic space. Let $\tau \in \{ fppf, \linebreak[0] {\acute{e}tale}, \linebreak[0] smooth, \linebreak[0] syntomic\} $. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces which is $\tau $-local on the target. Assume $\{ s : R \to U\} $ and $\{ t : R \to U\} $ are coverings for the $\tau $-topology. Let $W \subset U$ be the maximal open subspace such that $G_ W \to W$ has property $\mathcal{P}$. Then $W$ is $R$-invariant (see Groupoids in Spaces, Definition 77.18.1).

Proof. The existence and properties of the open $W \subset U$ are described in Descent on Spaces, Lemma 73.10.3. The morphism

\[ G \times _{U, t} R \longrightarrow R \times _{s, U} G, \quad (g, r) \longmapsto (r, r^{-1} \circ g \circ r) \]

is an isomorphism of algebraic spaces over $R$ (where $\circ $ denotes composition in the groupoid). Hence $s^{-1}(W) = t^{-1}(W)$ by the properties of $W$ proved in the aforementioned Descent on Spaces, Lemma 73.10.3. $\square$


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Your email address will not be published.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 06R4. Beware of the difference between the letter 'O' and the digit '0'.