Lemma 32.9.5. Let f : X \to S be a morphism of schemes. Assume
The morphism f is of locally of finite type.
The scheme X is quasi-compact and quasi-separated, and
The scheme S is quasi-separated.
Then X = \mathop{\mathrm{lim}}\nolimits X_ i where the X_ i \to S are of finite presentation, the X_ i are quasi-compact and quasi-separated, and the transition morphisms X_{i'} \to X_ i are closed immersions (which implies that X \to X_ i are closed immersions for all i).
Proof. By Lemma 32.9.3 there is a closed immersion X \to Y with Y \to S of finite presentation. Then Y is quasi-separated by Schemes, Lemma 26.21.12. Since X is quasi-compact, we may assume Y is quasi-compact by replacing Y with a quasi-compact open containing X. We see that X = \mathop{\mathrm{lim}}\nolimits X_ i with X_ i \to Y a closed immersion of finite presentation by Lemma 32.9.4. The morphisms X_ i \to S are of finite presentation by Morphisms, Lemma 29.21.3. \square
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