The Stacks project

Lemma 32.13.2. Let $f : X \to S$ be a proper morphism with $S$ quasi-compact and quasi-separated. Then $X = \mathop{\mathrm{lim}}\nolimits X_ i$ is a directed limit of schemes $X_ i$ proper and of finite presentation over $S$ such that all transition morphisms and the morphisms $X \to X_ i$ are closed immersions.

Proof. By Proposition 32.9.6 we can find a closed immersion $X \to Y$ with $Y$ separated and of finite presentation over $S$. By Lemma 32.12.1 we can find a diagram

\[ \xymatrix{ Y \ar[rd] & Y' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^ n_ S \ar[dl] \\ & S & } \]

where $Y' \to \mathbf{P}^ n_ S$ is an immersion, and $\pi : Y' \to Y$ is proper and surjective. By Lemma 32.9.4 we can write $X = \mathop{\mathrm{lim}}\nolimits X_ i$ with $X_ i \to Y$ a closed immersion of finite presentation. Denote $X'_ i \subset Y'$, resp. $X' \subset Y'$ the scheme theoretic inverse image of $X_ i \subset Y$, resp. $X \subset Y$. Then $\mathop{\mathrm{lim}}\nolimits X'_ i = X'$. Since $X' \to S$ is proper (Morphisms, Lemmas 29.41.4), we see that $X' \to \mathbf{P}^ n_ S$ is a closed immersion (Morphisms, Lemma 29.41.7). Hence for $i$ large enough we find that $X'_ i \to \mathbf{P}^ n_ S$ is a closed immersion by Lemma 32.4.20. Thus $X'_ i$ is proper over $S$. For such $i$ the morphism $X_ i \to S$ is proper by Morphisms, Lemma 29.41.9. $\square$

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