Lemma 68.8.7. Let $S$ be a scheme. Let $X$ be a quasi-separated algebraic space over $S$. Let $E \subset |X|$ be a subset. Then $E$ is étale locally constructible (Properties of Spaces, Definition 66.8.2) if and only if $E$ is a locally constructible subset of the topological space $|X|$ (Topology, Definition 5.15.1).
Proof. Assume $E \subset |X|$ is a locally constructible subset of the topological space $|X|$. Let $f : U \to X$ be an étale morphism where $U$ is a scheme. We have to show that $f^{-1}(E)$ is locally constructible in $U$. The question is local on $U$ and $X$, hence we may assume that $X$ is quasi-compact, $E \subset |X|$ is constructible, and $U$ is affine. In this case $U \to X$ is quasi-compact, hence $f : |U| \to |X|$ is quasi-compact. Observe that retrocompact opens of $|X|$, resp. $U$ are the same thing as quasi-compact opens of $|X|$, resp. $U$, see Topology, Lemma 5.27.1. Thus $f^{-1}(E)$ is constructible by Topology, Lemma 5.15.3.
Conversely, assume $E$ is étale locally constructible. We want to show that $E$ is locally constructible in the topological space $|X|$. The question is local on $X$, hence we may assume that $X$ is quasi-compact as well as quasi-separated. We will show that in this case $E$ is constructible in $|X|$. Choose open subspaces
and surjective étale morphisms $f_ p : V_ p \to U_ p$ inducing isomorphisms $f_ p^{-1}(T_ p) \to T_ p = U_ p \setminus U_{p + 1}$ where $V_ p$ is a quasi-compact separated scheme as in Lemma 68.8.6. By definition the inverse image $E_ p \subset V_ p$ of $E$ is locally constructible in $V_ p$. Then $E_ p$ is constructible in $V_ p$ by Properties, Lemma 28.2.5. Thus $E_ p \cap |f_ p^{-1}(T_ p)| = E \cap |T_ p|$ is constructible in $|T_ p|$ by Topology, Lemma 5.15.7 (observe that $V_ p \setminus f_ p^{-1}(T_ p)$ is quasi-compact as it is the inverse image of the quasi-compact space $U_{p + 1}$ by the quasi-compact morphism $f_ p$). Thus
is constructible by Topology, Lemma 5.15.14. Here we use that $|T_ p|$ is constructible in $|X|$ which is clear from what was said above. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)