Lemma 101.51.2. Let f : \mathcal{Y} \to \mathcal{X} be a morphism of algebraic stacks. Let x \in |\mathcal{X}| be a point. Assume
\mathcal{X} is decent or locally Noetherian (or both),
\mathcal{I}_\mathcal {X} \to \mathcal{X} is quasi-compact,
|f|(|\mathcal{Y}|) is contained in \{ x\} \subset |\mathcal{X}|, and
\mathcal{Y} is reduced.
Then f factors through the residual gerbe \mathcal{Z}_ x of \mathcal{X} at x (whose existence is guaranteed by Lemma 101.31.1 or 101.31.3).
Proof.
Let T = \overline{\{ x\} } \subset |\mathcal{X}| be the closure of x. By Properties of Stacks, Lemma 100.10.1 there exists a reduced closed substack \mathcal{X}' \subset \mathcal{X} such that T = |\mathcal{X}'|. By Properties of Stacks, Lemma 100.10.3 the morphism f factors through \mathcal{X}'. If \mathcal{X} is decent, then by Lemma 101.48.3 the stack \mathcal{X}' is decent. If \mathcal{X} is locally Noetherian, then \mathcal{X}' is locally Noetherian (details omitted). Note that \mathcal{I}_{\mathcal{X}'} \to \mathcal{X}' is the base change of \mathcal{I}_\mathcal {X} \to \mathcal{X} by Lemma 101.5.6 we see that \mathcal{I}_{\mathcal{X}'} \to \mathcal{X}' is quasi-compact by Lemma 101.7.3. This reduces us to the case discussed in the next paragraph.
Assume \mathcal{X} is reduced and x \in |\mathcal{X}| is a generic point. By Proposition 101.29.1 implies there exists a dense open substack \mathcal{U} \subset \mathcal{X}' which is a gerbe. Note that x \in |\mathcal{U}|. Repeating the arguments above we reduce to the case discussed in the next paragraph.
Assume \mathcal{X} \to X is a gerbe over the algebraic space X. If \mathcal{X} is decent, then by Lemmas 101.28.13 and 101.48.4 the space X is decent. If \mathcal{X} is locally Noetherian, then X is locally Noetherian by fppf descent (details omitted). Hence the corresponding result holds for X, see Decent Spaces, Lemma 68.13.10 or 68.13.9 (small detail omitted). Applying Lemma 101.51.1 we conclude that the result holds for \mathcal{X} as well.
\square
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