The Stacks project

Lemma 77.29.1. Let $B \to S$ be as in Section 77.3. Let $(U, R, s, t, c)$ be a groupoid in algebraic spaces over $B$. Let $G \to U$ be the stabilizer group algebraic space. The commutative diagram

\[ \xymatrix{ R \ar[d]^{\Delta _{R/U \times _ B U}} \ar[rrr]_{f \mapsto (f, s(f))} & & & R \times _{s, U} U \ar[d] \ar[r] & U \ar[d] \\ R \times _{(U \times _ B U)} R \ar[rrr]^{(f, g) \mapsto (f, f^{-1} \circ g)} & & & R \times _{s, U} G \ar[r] & G } \]

the two left horizontal arrows are isomorphisms and the right square is a fibre product square.

Proof. Omitted. Exercise in the definitions and the functorial point of view in algebraic geometry. $\square$


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Your email address will not be published.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 0454. Beware of the difference between the letter 'O' and the digit '0'.