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 $x$.
101.51 Residual gerbes
This section is the continuation of Properties of Stacks, Section 100.11.
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. $\square$
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. $\square$
Remark 101.51.3. We do not know whether Lemma 101.51.2 holds if we only assume $\mathcal{X}$ is locally Noetherian, i.e., we drop the assumption on the inertia being quasi-compact. In this case, if $x$ is a closed point, this is certainly true as follows from the following much simpler lemma.
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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