The Stacks project

Lemma 101.31.2. 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 101.21.3. 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 94.16.1) which is an equivalence because $U \to \mathcal{X}$ is surjective, flat, and locally of finite presentation, see Algebraic Stacks, Remark 94.16.3. 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 72.10.9 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 67.28.7 applies). Hence $\mathcal{Z} = [U'/R']$ is an algebraic stack by Criteria for Representability, Theorem 97.17.2. 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 78.25.1 and Properties of Stacks, Lemma 100.8.4. Since $\mathcal{Z} \to \mathcal{X}$ is a monomorphism we see that $|\mathcal{Z}| \to |\mathcal{X}|$ is injective, see Properties of Stacks, Lemma 100.8.5. By Properties of Stacks, Lemma 100.4.3 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 100.11.12. $\square$

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