The Stacks project

Lemma 76.41.2. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Assume $f$ separated of finite type, and $Y$ separated and quasi-compact. Then there exists a commutative diagram

\[ \xymatrix{ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & \mathbf{P}^ n_ Y \ar[ld] \\ & Y } \]

where $X' \to X$ is proper surjective morphism and the morphism $X' \to \mathbf{P}^ n_ Y$ is an immersion.

Proof. By Limits of Spaces, Proposition 70.11.7 we can find a closed immersion $X \to X_1$ where $X_1$ is separated and of finite presentation over $Y$. Clearly, if we prove the assertion for $X_1 \to Y$, then the result follows for $X$. Hence we may assume that $X$ is of finite presentation over $Y$.

We may and do replace the base scheme $S$ by $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Write $Y = \mathop{\mathrm{lim}}\nolimits _ i Y_ i$ as a directed limit of quasi-separated algebraic spaces of finite type over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Limits of Spaces, Proposition 70.8.1. By Limits of Spaces, Lemma 70.5.9 we may assume that $Y_ i$ is separated for all $i$. By Limits of Spaces, Lemma 70.7.1 we can find an index $i \in I$ and a scheme $X_ i \to Y_ i$ of finite presentation so that $X = Y \times _{Y_ i} X_ i$. By Limits of Spaces, Lemma 70.6.9 we may assume that $X_ i \to Y_ i$ is separated. Clearly, if we prove the assertion for $X_ i$ over $Y_ i$, then the assertion holds for $X$. The case $X_ i \to Y_ i$ is treated by Lemma 76.40.5. $\square$

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Your email address will not be published.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 089M. Beware of the difference between the letter 'O' and the digit '0'.