The Stacks project

Remark 99.10.5. Let $X$ be an algebraic stack. Let $T \subset |\mathcal{X}|$ be a locally closed subset. Let $\partial T$ be the boundary of $T$ in the topological space $|\mathcal{X}|$. In a formula

\[ \partial T = \overline{T} \setminus T. \]

Let $\mathcal{U} \subset \mathcal{X}$ be the open substack of $X$ with $|\mathcal{U}| = |\mathcal{X}| \setminus \partial T$, see Lemma 99.9.12. Let $\mathcal{Z}$ be the reduced closed substack of $\mathcal{U}$ with $|\mathcal{Z}| = T$ obtained by taking the reduced induced closed subspace structure, see Definition 99.10.4. By construction $\mathcal{Z} \to \mathcal{U}$ is a closed immersion of algebraic stacks and $\mathcal{U} \to \mathcal{X}$ is an open immersion, hence $\mathcal{Z} \to \mathcal{X}$ is an immersion of algebraic stacks by Lemma 99.9.3. Note that $\mathcal{Z}$ is a reduced algebraic stack and that $|\mathcal{Z}| = T$ as subsets of $|X|$. We sometimes say $\mathcal{Z}$ is the reduced induced substack structure on $T$.

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