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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