The Stacks project

95.30 Existence of residual gerbes

In this section we prove that residual gerbes (as defined in Properties of Stacks, Definition 94.11.8) exist on many algebraic stacks. First, here is the promised application of Proposition 95.28.1.

Lemma 95.30.1. 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 94.10.1 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 95.5.6. Hence $\mathcal{I}_{\mathcal{X}'} \to \mathcal{X}'$ is quasi-compact as a base change, see Lemma 95.7.3. Therefore Proposition 95.28.1 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 95.27.15. 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 95.30.2. 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 95.21.3. 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 88.16.1) which is an equivalence because $U \to \mathcal{X}$ is surjective, flat, and locally of finite presentation, see Algebraic Stacks, Remark 88.16.3. 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 66.10.9 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 61.28.7 applies). Hence $\mathcal{Z} = [U'/R']$ is an algebraic stack by Criteria for Representability, Theorem 91.17.2. 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 72.24.1 and Properties of Stacks, Lemma 94.8.4. Since $\mathcal{Z} \to \mathcal{X}$ is a monomorphism we see that $|\mathcal{Z}| \to |\mathcal{X}|$ is injective, see Properties of Stacks, Lemma 94.8.5. By Properties of Stacks, Lemma 94.4.3 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 94.11.11. $\square$


Comments (1)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 06UH. Beware of the difference between the letter 'O' and the digit '0'.