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.

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