The Stacks project

Lemma 32.12.1. Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a separated morphism of finite type. Then there exists an $n \geq 0$ and a diagram

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

where $X' \to \mathbf{P}^ n_ S$ is an immersion, and $\pi : X' \to X$ is proper and surjective.

Proof. By Proposition 32.9.6 we can find a closed immersion $X \to Y$ where $Y$ is separated and of finite presentation over $S$. Clearly, if we prove the assertion for $Y$, then the result follows for $X$. Hence we may assume that $X$ is of finite presentation over $S$.

Write $S = \mathop{\mathrm{lim}}\nolimits _ i S_ i$ as a directed limit of Noetherian schemes, see Proposition 32.5.4. By Lemma 32.10.1 we can find an index $i \in I$ and a scheme $X_ i \to S_ i$ of finite presentation so that $X = S \times _{S_ i} X_ i$. By Lemma 32.8.6 we may assume that $X_ i \to S_ i$ is separated. Clearly, if we prove the assertion for $X_ i$ over $S_ i$, then the assertion holds for $X$. The case $X_ i \to S_ i$ is treated by Cohomology of Schemes, Lemma 30.18.1. $\square$


Comments (0)


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Your email address will not be published.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.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0202. Beware of the difference between the letter 'O' and the digit '0'.