The Stacks project

Lemma 99.9.8. Let $(U, R, s, t, c)$ be a smooth groupoid in algebraic spaces. Let $\mathcal{X} = [U/R]$ be the associated algebraic stack, see Algebraic Stacks, Theorem 93.17.3. Let $Z \subset U$ be an $R$-invariant locally closed subspace. Then

\[ [Z/R_ Z] \longrightarrow [U/R] \]

is an immersion of algebraic stacks, where $R_ Z$ is the restriction of $R$ to $Z$. If $Z \subset U$ is open (resp. closed) then the morphism is an open (resp. closed) immersion of algebraic stacks.

Proof. Recall that by Groupoids in Spaces, Definition 77.18.1 (see also discussion following the definition) we have $R_ Z = s^{-1}(Z) = t^{-1}(Z)$ as locally closed subspaces of $R$. Hence the two morphisms $R_ Z \to Z$ are smooth as base changes of $s$ and $t$. Hence $(Z, R_ Z, s|_{R_ Z}, t|_{R_ Z}, c|_{R_ Z \times _{s, Z, t} R_ Z})$ is a smooth groupoid in algebraic spaces, and we see that $[Z/R_ Z]$ is an algebraic stack, see Algebraic Stacks, Theorem 93.17.3. The assumptions of Groupoids in Spaces, Lemma 77.25.3 are all satisfied and it follows that we have a $2$-fibre square

\[ \xymatrix{ Z \ar[d] \ar[r] & [Z/R_ Z] \ar[d] \\ U \ar[r] & [U/R] } \]

It follows from this and Lemma 99.3.1 that $[Z/R_ Z] \to [U/R]$ is representable by algebraic spaces, whereupon it follows from Lemma 99.3.3 that the right vertical arrow is an immersion (resp. closed immersion, resp. open immersion) if and only if the left vertical arrow is. $\square$


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