The Stacks project

Lemma 87.28.1. With assumptions and notation as in Theorem 87.27.4 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 87.23.5. 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 66.8.8. By Formal Spaces, Lemma 86.17.3 the base changes $W_ i \times _{X_{/T}} (X_ i)_{/T} \to (X_ i)_{/T}$ are quasi-compact. By Lemma 87.27.1 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

\[ U \amalg \coprod \nolimits _{j \in J'} V_ j \longrightarrow X' \]

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

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