The Stacks project

Lemma 32.4.17. Let $S$ be a scheme. Let $X = \mathop{\mathrm{lim}}\nolimits X_ i$ be a directed limit of schemes over $S$ with affine transition morphisms. Assume

  1. $S$ quasi-separated,

  2. $X_ i$ quasi-compact and quasi-separated,

  3. $X \to S$ separated.

Then $X_ i \to S$ is separated for all $i$ large enough.

Proof. Let $0 \in I$. Note that $I$ is nonempty as the limit is directed. As $X_0$ is quasi-compact we can find finitely many affine opens $U_1, \ldots , U_ n \subset S$ such that $X_0 \to S$ maps into $U_1 \cup \ldots \cup U_ n$. Denote $h_ i : X_ i \to S$ the structure morphism. It suffices to check that for some $i \geq 0$ the morphisms $h_ i^{-1}(U_ j) \to U_ j$ are separated for $j = 1, \ldots , n$. Since $S$ is quasi-separated the morphisms $U_ j \to S$ are quasi-compact. Hence $h_ i^{-1}(U_ j)$ is quasi-compact and quasi-separated. In this way we reduce to the case $S$ affine. In this case we have to show that $X_ i$ is separated and we know that $X$ is separated. Thus the lemma follows from Lemma 32.4.14. $\square$

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