Lemma 32.4.18. 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
S quasi-compact and quasi-separated,
X_ i quasi-compact and quasi-separated,
X \to S affine.
Then X_ i \to S is affine for i large enough.
Proof. Choose a finite affine open covering S = \bigcup _{j = 1, \ldots , n} V_ j. Denote f : X \to S and f_ i : X_ i \to S the structure morphisms. For each j the scheme f^{-1}(V_ j) = \mathop{\mathrm{lim}}\nolimits _ i f_ i^{-1}(V_ j) is affine (as a finite morphism is affine by definition). Hence by Lemma 32.4.13 there exists an i \in I such that each f_ i^{-1}(V_ j) is affine. In other words, f_ i : X_ i \to S is affine for i large enough, see Morphisms, Lemma 29.11.3. \square
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