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 (2)

Comment #5902 by Harry Gindi on

I think it's true here that the map π:X'→X can even be chosen to be finite presentation , surjective, and proper (it's a pullback along a closed immersion of such a map in the finite presentation case, which is itself a pullback of such a map in the Noetherian case). This is useful because if a map is proper and finite presentation, then pushforward of a constructible sheaf is constructible (as are all of the higher direct images). This might be useful to note in the statement (I'm writing something that makes use of this).

Comment #6103 by on

Yes, you are right. I formulated a remark with some additional properties you can assume in the situation of the lemma. I don't yet have a tag for it, but it should appear in the text as the remark following this lemma in a few days. Here is the commit.

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Your email address will not be published.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'.