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 $S$.

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