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

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. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)