Lemma 32.9.8 (09YY)—The Stacks project

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Table of contents
Part 2: Schemes
Chapter 32: Limits of Schemes
Section 32.9: Finite type closed in finite presentation
Lemma 32.9.8 (cite)

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Lemma 32.9.8. Let $f : X \to S$ be a morphism of schemes. Assume

$f$ is finite, and
$S$ quasi-compact and quasi-separated.

Then $X$ is a directed limit $X = \mathop{\mathrm{lim}}\nolimits X_ i$ where the transition maps are closed immersions and the objects $X_ i$ are finite and of finite presentation over $S$ .

Proof. We may write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ as in Lemma 32.9.5. Applying Lemma 32.4.19 we see that $X_ i \to S$ is finite for large enough $i$ . $\square$

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