The Stacks project

Lemma 32.9.5. Let $f : X \to S$ be a morphism of schemes. Assume

  1. The morphism $f$ is of locally of finite type.

  2. The scheme $X$ is quasi-compact and quasi-separated, and

  3. 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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