The Stacks project

65.29 Constructible sets

This section is the continuation of Properties of Spaces, Section 64.8.

Lemma 65.29.1. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $E \subset |Y|$ be a subset. If $E$ is étale locally constructible in $Y$, then $f^{-1}(E)$ is étale locally constructible in $X$.

Proof. Choose a scheme $V$ and a surjective étale morphism $\varphi : V \to Y$. Choose a scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$. Then $U \to X$ is surjective étale and the inverse image of $f^{-1}(E)$ in $U$ is the inverse image of $\varphi ^{-1}(E)$ by $U \to V$. Thus the lemma follows from the case of schemes for $U \to V$ (Morphisms, Lemma 29.22.1) and the definition (Properties of Spaces, Definition 64.8.2). $\square$

Theorem 65.29.2 (Chevalley's Theorem). Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ is quasi-compact and locally of finite presentation. Then the image of every étale locally constructible subset of $|X|$ is an étale locally constructible subset of $|Y|$.

Proof. Let $E \subset |X|$ be étale locally constructible. Let $V \to Y$ be an étale morphism with $V$ affine. It suffices to show that the inverse image of $f(E)$ in $V$ is constructible, see Properties of Spaces, Definition 64.8.2. Since $f$ is quasi-compact $V \times _ Y X$ is a quasi-compact algebraic space. Choose an affine scheme $U$ and a surjective étale morphism $U \to V \times _ Y X$ (Properties of Spaces, Lemma 64.6.3). By Properties of Spaces, Lemma 64.4.3 the inverse image of $f(E)$ in $V$ is the image under $U \to V$ of the inverse image of $E$ in $U$. Thus the result follows from the case of schemes, see Morphisms, Lemma 29.22.2. $\square$


Comments (0)


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Your email address will not be published.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