## 100.31 Existence of residual gerbes

In this section we prove that residual gerbes (as defined in Properties of Stacks, Definition 99.11.8) exist on many algebraic stacks. First, here is the promised application of Proposition 100.29.1.

Lemma 100.31.1. Let $\mathcal{X}$ be an algebraic stack such that $\mathcal{I}_\mathcal {X} \to \mathcal{X}$ is quasi-compact. Then the residual gerbe of $\mathcal{X}$ at $x$ exists for every $x \in |\mathcal{X}|$.

Proof. Let $T = \overline{\{ x\} } \subset |\mathcal{X}|$ be the closure of $x$. By Properties of Stacks, Lemma 99.10.1 there exists a reduced closed substack $\mathcal{X}' \subset \mathcal{X}$ such that $T = |\mathcal{X}'|$. Note that $\mathcal{I}_{\mathcal{X}'} = \mathcal{I}_\mathcal {X} \times _\mathcal {X} \mathcal{X}'$ by Lemma 100.5.6. Hence $\mathcal{I}_{\mathcal{X}'} \to \mathcal{X}'$ is quasi-compact as a base change, see Lemma 100.7.3. Therefore Proposition 100.29.1 implies there exists a dense open substack $\mathcal{U} \subset \mathcal{X}'$ which is a gerbe. Note that $x \in |\mathcal{U}|$ because $\{ x\} \subset T$ is a dense subset too. Hence a residual gerbe $\mathcal{Z}_ x \subset \mathcal{U}$ of $\mathcal{U}$ at $x$ exists by Lemma 100.28.15. It is immediate from the definitions that $\mathcal{Z}_ x \to \mathcal{X}$ is a residual gerbe of $\mathcal{X}$ at $x$. $\square$

If the stack is quasi-DM then residual gerbes exist too. In particular, residual gerbes always exist for Deligne-Mumford stacks.

Lemma 100.31.2. Let $\mathcal{X}$ be a quasi-DM algebraic stack. Then the residual gerbe of $\mathcal{X}$ at $x$ exists for every $x \in |\mathcal{X}|$.

Proof. Choose a scheme $U$ and a surjective, flat, locally finite presented, and locally quasi-finite morphism $U \to \mathcal{X}$, see Theorem 100.21.3. Set $R = U \times _\mathcal {X} U$. The projections $s, t : R \to U$ are surjective, flat, locally of finite presentation, and locally quasi-finite as base changes of the morphism $U \to \mathcal{X}$. There is a canonical morphism $[U/R] \to \mathcal{X}$ (see Algebraic Stacks, Lemma 93.16.1) which is an equivalence because $U \to \mathcal{X}$ is surjective, flat, and locally of finite presentation, see Algebraic Stacks, Remark 93.16.3. Thus we may assume that $\mathcal{X} = [U/R]$ where $(U, R, s, t, c)$ is a groupoid in algebraic spaces such that $s, t : R \to U$ are surjective, flat, locally of finite presentation, and locally quasi-finite. Set

$U' = \coprod \nolimits _{u \in U\text{ lying over }x} \mathop{\mathrm{Spec}}(\kappa (u)).$

The canonical morphism $U' \to U$ is a monomorphism. Let

$R' = U' \times _\mathcal {X} U' = R \times _{(U \times U)} (U' \times U')$

Because $U' \to U$ is a monomorphism we see that both projections $s', t' : R' \to U'$ factor as a monomorphism followed by a locally quasi-finite morphism. Hence, as $U'$ is a disjoint union of spectra of fields, using Spaces over Fields, Lemma 71.10.9 we conclude that the morphisms $s', t' : R' \to U'$ are locally quasi-finite. Again since $U'$ is a disjoint union of spectra of fields, the morphisms $s', t'$ are also flat. Finally, $s', t'$ locally quasi-finite implies $s', t'$ locally of finite type, hence $s', t'$ locally of finite presentation (because $U'$ is a disjoint union of spectra of fields in particular locally Noetherian, so that Morphisms of Spaces, Lemma 66.28.7 applies). Hence $\mathcal{Z} = [U'/R']$ is an algebraic stack by Criteria for Representability, Theorem 96.17.2. As $R'$ is the restriction of $R$ by $U' \to U$ we see $\mathcal{Z} \to \mathcal{X}$ is a monomorphism by Groupoids in Spaces, Lemma 77.25.1 and Properties of Stacks, Lemma 99.8.4. Since $\mathcal{Z} \to \mathcal{X}$ is a monomorphism we see that $|\mathcal{Z}| \to |\mathcal{X}|$ is injective, see Properties of Stacks, Lemma 99.8.5. By Properties of Stacks, Lemma 99.4.3 we see that

$|U'| = |\mathcal{Z} \times _\mathcal {X} U'| \longrightarrow |\mathcal{Z}| \times _{|\mathcal{X}|} |U'|$

is surjective which implies (by our choice of $U'$) that $|\mathcal{Z}| \to |\mathcal{X}|$ has image $\{ x\}$. We conclude that $|\mathcal{Z}|$ is a singleton. Finally, by construction $U'$ is locally Noetherian and reduced, i.e., $\mathcal{Z}$ is reduced and locally Noetherian. This means that the essential image of $\mathcal{Z} \to \mathcal{X}$ is the residual gerbe of $\mathcal{X}$ at $x$, see Properties of Stacks, Lemma 99.11.11. $\square$

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