The Stacks project

Lemma 100.18.7. Let $\mathcal{X}$ be an algebraic stack. Let $x \in |\mathcal{X}|$. The following are equivalent:

  1. $x$ is a finite type point,

  2. there exists an algebraic stack $\mathcal{Z}$ whose underlying topological space $|\mathcal{Z}|$ is a singleton, and a morphism $f : \mathcal{Z} \to \mathcal{X}$ which is locally of finite type such that $\{ x\} = |f|(|\mathcal{Z}|)$, and

  3. the residual gerbe $\mathcal{Z}_ x$ of $\mathcal{X}$ at $x$ exists and the inclusion morphism $\mathcal{Z}_ x \to \mathcal{X}$ is locally of finite type.

Proof. (All of the morphisms occurring in this paragraph are representable by algebraic spaces, hence the conventions and results of Properties of Stacks, Section 99.3 are applicable.) Assume $x$ is a finite type point. Choose an affine scheme $U$, a closed point $u \in U$, and a smooth morphism $\varphi : U \to \mathcal{X}$ with $\varphi (u) = x$, see Lemma 100.18.3. Set $u = \mathop{\mathrm{Spec}}(\kappa (u))$ as usual. Set $R = u \times _\mathcal {X} u$ so that we obtain a groupoid in algebraic spaces $(u, R, s, t, c)$, see Algebraic Stacks, Lemma 93.16.1. The projection morphisms $R \to u$ are the compositions

\[ R = u \times _\mathcal {X} u \to u \times _\mathcal {X} U \to u \times _\mathcal {X} X = u \]

where the first arrow is of finite type (a base change of the closed immersion of schemes $u \to U$) and the second arrow is smooth (a base change of the smooth morphism $U \to \mathcal{X}$). Hence $s, t : R \to u$ are locally of finite type (as compositions, see Morphisms of Spaces, Lemma 66.23.2). Since $u$ is the spectrum of a field, it follows that $s, t$ are flat and locally of finite presentation (by Morphisms of Spaces, Lemma 66.28.7). We see that $\mathcal{Z} = [u/R]$ is an algebraic stack by Criteria for Representability, Theorem 96.17.2. By Algebraic Stacks, Lemma 93.16.1 we obtain a canonical morphism

\[ f : \mathcal{Z} \longrightarrow \mathcal{X} \]

which is fully faithful. Hence this morphism is representable by algebraic spaces, see Algebraic Stacks, Lemma 93.15.2 and a monomorphism, see Properties of Stacks, Lemma 99.8.4. It follows that the residual gerbe $\mathcal{Z}_ x \subset \mathcal{X}$ of $\mathcal{X}$ at $x$ exists and that $f$ factors through an equivalence $\mathcal{Z} \to \mathcal{Z}_ x$, see Properties of Stacks, Lemma 99.11.12. By construction the diagram

\[ \xymatrix{ u \ar[d] \ar[r] & U \ar[d] \\ \mathcal{Z} \ar[r]^ f & \mathcal{X} } \]

is commutative. By Criteria for Representability, Lemma 96.17.1 the left vertical arrow is surjective, flat, and locally of finite presentation. Consider

\[ \xymatrix{ u \times _\mathcal {X} U \ar[d] \ar[r] & \mathcal{Z} \times _\mathcal {X} U \ar[r] \ar[d] & U \ar[d] \\ u \ar[r] & \mathcal{Z} \ar[r]^ f & \mathcal{X} } \]

As $u \to \mathcal{X}$ is locally of finite type, we see that the base change $u \times _\mathcal {X} U \to U$ is locally of finite type. Moreover, $u \times _\mathcal {X} U \to \mathcal{Z} \times _\mathcal {X} U$ is surjective, flat, and locally of finite presentation as a base change of $u \to \mathcal{Z}$. Thus $\{ u \times _\mathcal {X} U \to \mathcal{Z} \times _\mathcal {X} U\} $ is an fppf covering of algebraic spaces, and we conclude that $\mathcal{Z} \times _\mathcal {X} U \to U$ is locally of finite type by Descent on Spaces, Lemma 73.16.1. By definition this means that $f$ is locally of finite type (because the vertical arrow $\mathcal{Z} \times _\mathcal {X} U \to \mathcal{Z}$ is smooth as a base change of $U \to \mathcal{X}$ and surjective as $\mathcal{Z}$ has only one point). Since $\mathcal{Z} = \mathcal{Z}_ x$ we see that (3) holds.

It is clear that (3) implies (2). If (2) holds then $x$ is a finite type point of $\mathcal{X}$ by Lemma 100.18.4 and Lemma 100.18.6 to see that $\mathcal{Z}_{\text{ft-pts}}$ is nonempty, i.e., the unique point of $\mathcal{Z}$ is a finite type point of $\mathcal{Z}$. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 100.18: Points of finite type

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 06G3. Beware of the difference between the letter 'O' and the digit '0'.