Lemma 101.51.1. Let \pi : \mathcal{X} \to \mathcal{Y} be a morphism of algebraic stacks. Let x \in |\mathcal{X}| with image y \in |\mathcal{Y}|. Assume the residual gerbe \mathcal{Z}_ y \subset \mathcal{Y} of \mathcal{Y} at y exists and that \mathcal{X} is a gerbe over \mathcal{Y}. Then \mathcal{Z}_ x = \mathcal{Z}_ y \times _\mathcal {Y} \mathcal{X} is the residual gerbe of \mathcal{X} at x.
Proof. The morphism \mathcal{Z}_ x \to \mathcal{X} is a monomorphism as the base change of the monomorphism \mathcal{Z}_ y \to \mathcal{Y}. The morphism \pi is a universal homeomorphism by Lemma 101.28.13 and hence |\mathcal{Z}_ x| = \{ x\} . Finally, the morphism \mathcal{Z}_ x \to \mathcal{Z}_ y is smooth as a base change of the smooth morphism \pi , see Lemma 101.33.8. Hence as \mathcal{Z}_ y is reduced and locally Noetherian, so is \mathcal{Z}_ x (details omitted). Thus \mathcal{Z}_ x is the residual gerbe of \mathcal{X} at x by Properties of Stacks, Definition 100.11.8. \square
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