Lemma 86.24.1. With assumptions and notation as in Theorem 86.23.4 let $f : X' \to X$ correspond to $g : W \to X_{/T}$. Then $f$ is quasi-compact if and only if $g$ is quasi-compact.

**Proof.**
If $f$ is quasi-compact, then $g$ is quasi-compact by Lemma 86.20.4. Conversely, assume $g$ is quasi-compact. Choose an étale covering $\{ X_ i \to X\} $ with $X_ i$ affine. It suffices to prove that the base change $X' \times _ X X_ i \to X_ i$ is quasi-compact, see Morphisms of Spaces, Lemma 65.8.8. By Formal Spaces, Lemma 85.13.3 the base changes $W_ i \times _{X_{/T}} (X_ i)_{/T} \to (X_ i)_{/T}$ are quasi-compact. By Lemma 86.23.1 we reduce to the case described in the next paragraph.

Assume $X$ is affine and $g : W \to X_{/T}$ quasi-compact. We have to show that $X'$ is quasi-compact. Let $V \to X'$ be a surjective étale morphism where $V = \coprod _{j \in J} V_ j$ is a disjoint union of affines. Then $V_{/T} \to X'_{/T} = W$ is a surjective étale morphism. Since $W$ is quasi-compact, then we can find a finite subset $J' \subset J$ such that $\coprod _{j \in J'} (V_ j)_{/T} \to W$ is surjective. Then it follows that

is surjective (and hence $X'$ is quasi-compact). Namely, we have $|X'| = |U| \amalg |W_{red}|$ as $X'_{/T} = W$. $\square$

## 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Your email address will not be published.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.

## Comments (0)