The Stacks project

64.8 Constructible sets

Lemma 64.8.1. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \subset |X|$ be a subset. The following are equivalent

  1. for every étale morphism $U \to X$ where $U$ is a scheme the inverse image of $E$ in $U$ is a locally constructible subset of $U$,

  2. for every étale morphism $U \to X$ where $U$ is an affine scheme the inverse image of $E$ in $U$ is a constructible subset of $U$,

  3. for some surjective étale morphism $U \to X$ where $U$ is a scheme the inverse image of $E$ in $U$ is a locally constructible subset of $U$.

Proof. By Properties, Lemma 28.2.1 we see that (1) and (2) are equivalent. It is immediate that (1) implies (3). Thus we assume we have a surjective étale morphism $\varphi : U \to X$ where $U$ is a scheme such that $\varphi ^{-1}(E)$ is locally constructible. Let $\varphi ' : U' \to X$ be another étale morphism where $U'$ is a scheme. Then we have

\[ E'' = \text{pr}_1^{-1}(\varphi ^{-1}(E)) = \text{pr}_2^{-1}((\varphi ')^{-1}(E)) \]

where $\text{pr}_1 : U \times _ X U' \to U$ and $\text{pr}_2 : U \times _ X U' \to U'$ are the projections. By Morphisms, Lemma 29.21.1 we see that $E''$ is locally constructible in $U \times _ X U'$. Let $W' \subset U'$ be an affine open. Since $\text{pr}_2$ is étale and hence open, we can choose a quasi-compact open $W'' \subset U \times _ X U'$ with $\text{pr}_2(W'') = W'$. Then $\text{pr}_2|_{W''} : W'' \to W'$ is quasi-compact. We have $W \cap (\varphi ')^{-1}(E) = \text{pr}_2(E'' \cap W'')$ as $\varphi $ is surjective, see Lemma 64.4.3. Thus $W \cap (\varphi ')^{-1}(E) = \text{pr}_2(E'' \cap W'')$ is locally constructible by Morphisms, Theorem 29.21.3 as desired. $\square$

Definition 64.8.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \subset |X|$ be a subset. We say $E$ is étale locally constructible if the equivalent conditions of Lemma 64.8.1 are satisfied.

Of course, if $X$ is representable, i.e., $X$ is a scheme, then this just means $E$ is a locally constructible subset of the underlying topological space.


Comments (1)

Comment #5096 by Klaus Mattis on

In the proof of Lemma 0ECT, W is never defined. I guess the two "W"s in te last two sentences should actually be W'.


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Your email address will not be published.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