The Stacks project

Lemma 101.51.4. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Let $x \in |\mathcal{X}|$ with residual gerbe $\mathcal{Z}_ x \subset \mathcal{X}$ (Lemma 101.31.3). Then $x$ is a closed point of $|\mathcal{X}|$ if and only if the morphism $\mathcal{Z}_ x \to \mathcal{X}$ is a closed immersion.

Proof. If $\mathcal{Z}_ x \to \mathcal{X}$ is a closed immersion, then $x$ is a closed point of $|\mathcal{X}|$, see for example Lemma 101.37.4. Conversely, assume $x$ is a closed point of $|\mathcal{X}|$. Let $\mathcal{Z} \subset \mathcal{X}$ be the reduced closed substack with $|Z| = \{ x\} $ (Properties of Stacks, Lemma 100.10.1). Then $\mathcal{Z}$ is a locally Noetherian algebraic stack by Lemmas 101.17.4 and 101.17.5. Since also $\mathcal{Z}$ is reduced and $|\mathcal{Z}| = \{ x\} $ it follows that $\mathcal{Z} = \mathcal{Z}_ x$ is the residual gerbe by definition. $\square$

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