Lemma 67.25.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. The following are equivalent:

1. $x$ is a finite type point,

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

3. there exists an algebraic space $Z$ and a morphism $f : Z \to X$ with the following properties:

1. there is a surjective étale morphism $z : \mathop{\mathrm{Spec}}(k) \to Z$ where $k$ is a field,

2. $f$ is locally of finite type,

3. $f$ is a monomorphism, and

4. $x = f(z)$.

Proof. Assume $x$ is a finite type point. Choose an affine scheme $U$, a closed point $u \in U$, and an étale morphism $\varphi : U \to X$ with $\varphi (u) = x$, see Lemma 67.25.3. Set $u = \mathop{\mathrm{Spec}}(\kappa (u))$ as usual. The projection morphisms $u \times _ X u \to u$ are the compositions

$u \times _ X u \to u \times _ X U \to u \times _ X X = u$

where the first arrow is a closed immersion (a base change of $u \to U$) and the second arrow is étale (a base change of the étale morphism $U \to X$). Hence $u \times _ X U$ is a disjoint union of spectra of finite separable extensions of $k$ (see Morphisms, Lemma 29.36.7) and therefore the closed subscheme $u \times _ X u$ is a disjoint union of finite separable extension of $k$, i.e., $u \times _ X u \to u$ is étale. By Spaces, Theorem 65.10.5 we see that $Z = u/u \times _ X u$ is an algebraic space. By construction the diagram

$\xymatrix{ u \ar[d] \ar[r] & U \ar[d] \\ Z \ar[r] & X }$

is commutative with étale vertical arrows. Hence $Z \to X$ is locally of finite type (see Lemma 67.23.4). By construction the morphism $Z \to X$ is a monomorphism and the image of $z$ is $x$. Thus (3) holds.

It is clear that (3) implies (2). If (2) holds then $x$ is a finite type point of $X$ by Lemma 67.25.4 (and Lemma 67.25.6 to see that $Z_{\text{ft-pts}}$ is nonempty, i.e., the unique point of $Z$ is a finite type point of $Z$). $\square$

## 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Your email address will not be published.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 06EL. Beware of the difference between the letter 'O' and the digit '0'.