The Stacks project

Lemma 67.49.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The following are equivalent

  1. there is a surjective étale morphism $U \to X$ where $U$ is a scheme such that every quasi-compact open of $U$ has finitely many irreducible components,

  2. for every scheme $U$ and every étale morphism $U \to X$ every quasi-compact open of $U$ has finitely many irreducible components,

  3. for every quasi-compact algebraic space $Y$ étale over $X$ the set of codimension $0$ points of $Y$ (Properties of Spaces, Definition 66.10.2) is finite, and

  4. for every quasi-compact algebraic space $Y$ étale over $X$ the space $|Y|$ has finitely many irreducible components.

If $X$ is representable this means that every quasi-compact open of $X$ has finitely many irreducible components.

Proof. The equivalence of (1) and (2) and the final statement follow from Descent, Lemma 35.16.3 and Properties of Spaces, Lemma 66.7.1. It is clear that (4) implies (1) and (2) by considering only those $Y$ which are schemes. Similarly, (3) implies (1) and (2) since for a scheme the codimension $0$ points are the generic points of its irreducible components, see for example Properties of Spaces, Lemma 66.11.1.

Conversely, assume (2) and let $Y \to X$ be an étale morphism of algebraic spaces with $Y$ quasi-compact. Then we can choose an affine scheme $V$ and a surjective étale morphism $V \to Y$ (Properties of Spaces, Lemma 66.6.3). Since $V$ has finitely many irreducible components by (2) and since $|V| \to |Y|$ is surjective and continuous, we conclude that $|Y|$ has finitely many irreducible components by Topology, Lemma 5.8.5. Thus (4) holds. Similarly, by Properties of Spaces, Lemma 66.11.1 the images of the generic points of the irreducible components of $V$ are the codimension $0$ points of $Y$ and we conclude that there are finitely many, i.e., (3) holds. $\square$

Comments (2)

Comment #7686 by Anonymous on

In (1), the phrase "is as scheme" should be "is a scheme".

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Your email address will not be published.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 0BB1. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 67.49.1 as pdf