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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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