The Stacks project

Lemma 101.31.3. Let $\mathcal{X}$ be a locally Noetherian algebraic stack. Then the residual gerbe of $\mathcal{X}$ at $x$ exists for every $x \in |\mathcal{X}|$.

Proof. Choose an affine scheme $U$ and a smooth morphism $U \to \mathcal{X}$ such that $x$ is in the image of the open continuous map $|U| \to |\mathcal{X}|$. We may and do replace $\mathcal{X}$ with the open substack corresponding to the image of $|U| \to |\mathcal{X}|$, see Properties of Stacks, Lemma 100.9.12. Thus we may assume $\mathcal{X} = [U/R]$ for a smooth groupoid $(U, R, s, t, c)$ in algebraic spaces where $U$ is a Noetherian affine scheme, see Algebraic Stacks, Section 94.16.

Let $E \subset |U|$ be the inverse image of $\{ x\} \subset |\mathcal{X}|$. Of course $E \not= \emptyset $. Since $|U|$ is a Noetherian topological space, we can choose an element $u \in E$ such that $\overline{\{ u\} } \cap E = \{ u\} $. As usual, we think of $u = \mathop{\mathrm{Spec}}(\kappa (u))$ as the spectrum of its residue field. Let us write

\[ F = u \times _{U, t} R = u \times _\mathcal {X} U \quad \text{and}\quad R' = (u \times u) \times _{(U \times U), (t, s)} R = u \times _\mathcal {X} u \]

Furthermore, denote $Z = \overline{\{ u\} } \subset U$ with the reduced induced scheme structure. Denote $p : F \to U$ the morphism induced by the second projection (using $s : R \to U$ in the first fibre product description of $F$). Then $E$ is the set theoretic image of $p$. The morphism $R' \to F$ is a monomorphism which factors through the inverse image $p^{-1}(Z)$ of $Z$. This inverse image $p^{-1}(Z) \subset F$ is a closed subscheme and the restriction $p|_{p^{-1}(Z)} : p^{-1}(Z) \to Z$ has image set theoretically contained in $\{ u\} \subset Z$ by our careful choice of $u \in E$ above. Since $u = \mathop{\mathrm{lim}}\nolimits W$ where the limit is over the nonempty affine open subschemes of the irreducible reduced scheme $Z$, we conclude that the morphism $p|_{p^{-1}(Z)} : p^{-1}(Z) \to Z$ factors through the morphism $u \to Z$. Clearly this implies that $R' = p^{-1}(Z)$. In particular the morphism $t' : R' \to u$ is locally of finite presentation as the composition of the closed immersion $p^{-1}(Z) \to F$ of locally Noetherian algebraic spaces with the smooth morphism $\text{pr}_1 : F \to u$; use Morphisms of Spaces, Lemmas 67.23.5, 67.28.12, and 67.28.2. Hence the restriction $(u, R', s', t', c')$ of $(U, R, s, t, c)$ by $u \to U$ is a groupoid in algebraic spaces where $s'$ and $t'$ are flat and locally of finite presentation. Therefore $\mathcal{Z} = [u/R']$ 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. Then $\mathcal{Z}$ is (isomorphic to) the residual gerbe by the material in Properties of Stacks, Section 100.11. $\square$

Comments (0)

There are also:

  • 1 comment(s) on Section 101.31: Existence of residual gerbes

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 0H22. Beware of the difference between the letter 'O' and the digit '0'.