The Stacks project

Lemma 86.24.2. 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-separated if and only if $g$ is so.

Proof. If $f$ is quasi-separated, then $g$ is quasi-separated by Lemma 86.20.6. Conversely, assume $g$ is quasi-separated. We have to show that $f$ is quasi-separated. Exactly as in the proof of Lemma 86.24.1 we may check this over the members of a ├ętale covering of $X$ by affine schemes using Morphisms of Spaces, Lemma 65.4.12 and Formal Spaces, Lemma 85.25.5. Thus we may and do assume $X$ is affine.

Let $V \to X'$ be a surjective ├ętale morphism where $V = \coprod _{j \in J} V_ j$ is a disjoint union of affines. To show that $X'$ is quasi-separated, it suffices to show that $V_ j \times _{X'} V_{j'}$ is quasi-compact for all $j, j' \in J$. Since $W$ is quasi-separated the fibre products $(V_ j \times _ Y V_{j'})_{/T} = (V_ j)_{/T} \times _{X'_{/T}} (V_{j'})_{/T}$ are quasi-compact for all $j, j' \in J$. Since $X$ is Noetherian affine and $U' \to U$ is an isomorphism, we see that

\[ (V_ j \times _{X'} V_{j'}) \times _ X U = (V_ j \times _ X V_{j'}) \times _ X U \]

is quasi-compact. Hence we conclude by the equality

\[ |V_ j \times _{X'} V_{j'}| = |(V_ j \times _{X'} V_{j'}) \times _ X U| \amalg |(V_ j \times _{X'} V_{j'})_{/T, red}| \]

and the fact that a formal algebraic space is quasi-compact if and only if its associated reduced algebraic space is so. $\square$


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