The Stacks project

Lemma 39.13.2. Let $S$ be a scheme. Let $(U, R, s, t, c)$ be a groupoid scheme over $S$. Let $p \in S$ be a point, and let $u \in U$ be a point lying over $p$. Assume assumptions (1) – (6) of Lemma 39.13.1 hold as well as

  1. $j : R \to U \times _ S U$ is universally closed1.

Then we can choose $(S', p') \to (S, p)$ and decompositions $S' \times _ S U = U' \amalg W$ and $S' \times _ S R = R' \amalg W'$ and $u' \in U'$ such that (a) – (g) of Lemma 39.13.1 hold as well as

  1. $R'$ is the restriction of $S' \times _ S R$ to $U'$.

Proof. We apply Lemma 39.13.1 for the groupoid $(U, R, s, t, c)$ over the scheme $S$ with points $p$ and $u$. Hence we get an étale neighbourhood $(S', p') \to (S, p)$ and disjoint union decompositions

\[ S' \times _ S U = U' \amalg W, \quad S' \times _ S R = R' \amalg W' \]

and $u' \in U'$ satisfying conclusions (a), (b), (c), (d), (e), (f), and (g). We may shrink $S'$ to a smaller neighbourhood of $p'$ without affecting the conclusions (a) – (g). We will show that for a suitable shrinking conclusion (h) holds as well. Let us denote $j'$ the base change of $j$ to $S'$. By conclusion (e) it is clear that

\[ j'^{-1}(U' \times _{S'} U') = R' \amalg Rest \]

for some open and closed $Rest$ piece. Since $U' \to S'$ is finite by conclusion (d) we see that $U' \times _{S'} U'$ is finite over $S'$. Since $j$ is universally closed, also $j'$ is universally closed, and hence $j'|_{Rest}$ is universally closed too. By conclusions (b) and (c) we see that the fibre of

\[ (U' \times _{S'} U' \to S') \circ j'|_{Rest} : Rest \longrightarrow S' \]

over $p'$ is empty. Hence, since $Rest \to S'$ is closed as a composition of closed morphisms, after replacing $S'$ by $S' \setminus \mathop{\mathrm{Im}}(Rest \to S')$, we may assume that $Rest = \emptyset $. And this is exactly the condition that $R'$ is the restriction of $S' \times _ S R$ to the open subscheme $U' \subset S' \times _ S U$, see Groupoids, Lemma 38.18.3 and its proof. $\square$

[1] In view of the other conditions this is equivalent to requiring $j$ to be proper.

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Your email address will not be published.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 03X5. Beware of the difference between the letter 'O' and the digit '0'.