The Stacks project

Lemma 100.9.7. Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces. Let $i : \mathcal{Z} \to [U/R]$ be an immersion. Then there exists an $R$-invariant locally closed subspace $Z \subset U$ and a presentation $[Z/R_ Z] \to \mathcal{Z}$ where $R_ Z$ is the restriction of $R$ to $Z$ such that

\[ \xymatrix{ [Z/R_ Z] \ar[dr] \ar[rr] & & \mathcal{Z} \ar[ld]^ i \\ & [U/R] } \]

is $2$-commutative. If $i$ is a closed (resp. open) immersion then $Z$ is a closed (resp. open) subspace of $U$.

Proof. By Lemma 100.3.6 we get a commutative diagram

\[ \xymatrix{ [U'/R'] \ar[dr] \ar[rr] & & \mathcal{Z} \ar[ld] \\ & [U/R] } \]

where $U' = \mathcal{Z} \times _{[U/R]} U$ and $R' = \mathcal{Z} \times _{[U/R]} R$. Since $\mathcal{Z} \to [U/R]$ is an immersion we see that $U' \to U$ is an immersion of algebraic spaces. Let $Z \subset U$ be the locally closed subspace such that $U' \to U$ factors through $Z$ and induces an isomorphism $U' \to Z$. It is clear from the construction of $R'$ that $R' = U' \times _{U, t} R = R \times _{s, U} U'$. This implies that $Z \cong U'$ is $R$-invariant and that the image of $R' \to R$ identifies $R'$ with the restriction $R_ Z = s^{-1}(Z) = t^{-1}(Z)$ of $R$ to $Z$. Hence the lemma holds. $\square$

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