The Stacks project

[IV Theorem 3.1, Kn]

Lemma 76.40.5 (Chow's lemma). 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 Noetherian. 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 a $U$-admissible blowup for some dense open $U \subset X$ and the morphism $X' \to \mathbf{P}^ n_ Y$ is an immersion.

Proof. In this first paragraph of the proof we reduce the lemma to the case where $Y$ is of finite type over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. We may and do replace the base scheme $S$ by $\mathop{\mathrm{Spec}}(\mathbf{Z})$. We can write $Y = \mathop{\mathrm{lim}}\nolimits Y_ i$ as a directed limit of separated algebraic spaces of finite type over $\mathop{\mathrm{Spec}}(\mathbf{Z})$, see Limits of Spaces, Proposition 70.8.1 and Lemma 70.5.9. For all $i$ sufficiently large we can find a separated finite type morphism $X_ i \to Y_ i$ such that $X = Y \times _{Y_ i} X_ i$, see Limits of Spaces, Lemmas 70.7.1 and 70.6.9. Let $\eta _1, \ldots , \eta _ n$ be the generic points of the irreducible components of $|X|$ ($X$ is Noetherian as a finite type separated algebraic space over the Noetherian algebraic space $Y$ and therefore $|X|$ is a Noetherian topological space). By Limits of Spaces, Lemma 70.5.2 we find that the images of $\eta _1, \ldots , \eta _ n$ in $|X_ i|$ are distinct for $i$ large enough. We may replace $X_ i$ by the scheme theoretic image of the (quasi-compact, in fact affine) morphism $X \to X_ i$. After this replacement we see that the images of $\eta _1, \ldots , \eta _ n$ in $|X_ i|$ are the generic points of the irreducible components of $|X_ i|$, see Morphisms of Spaces, Lemma 67.16.3. Having said this, suppose we can find a diagram

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

where $X_ i' \to X_ i$ is a $U_ i$-admissible blowup for some dense open $U_ i \subset X_ i$ and the morphism $X_ i' \to \mathbf{P}^ n_{Y_ i}$ is an immersion. Then the strict transform $X' \to X$ of $X$ relative to $X_ i' \to X_ i$ is a $U$-admissible blowing up where $U \subset X$ is the inverse image of $U_ i$ in $X$. Because of our carefully chosen index $i$ it follows that $\eta _1, \ldots , \eta _ n \in |U|$ and $U \subset X$ is dense. Moreover, $X' \to \mathbf{P}^ n_ Y$ is an immersion as $X'$ is closed in $X_ i' \times _{X_ i} X = X_ i' \times _{Y_ i} Y$ which comes with an immersion into $\mathbf{P}^ n_ Y$. Thus we have reduced to the situation of the following paragraph.

Assume that $Y$ is separated of finite type over $\mathop{\mathrm{Spec}}(\mathbf{Z})$. Then $X \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is separated of finite type as well. We apply Lemma 76.40.3 to $X \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ to find a dense open subspace $U \subset X$ and a commutative diagram

\[ \xymatrix{ & U \ar[ld] \ar[d] \ar[rd] \ar[rrd] \\ X \ar[rd] & X' \ar[l] \ar[d] \ar[r] & Z' \ar[ld] \ar[r] & Z \ar[ld] \\ & \mathop{\mathrm{Spec}}(\mathbf{Z}) & \mathbf{P}^ n_\mathbf {Z} \ar[l] } \]

with all the properties listed in the lemma. Note that $Z$ has an ample invertible sheaf, namely $\mathcal{O}_{\mathbf{P}^ n}(1)|_ Z$. Hence $Z' \to Z$ is a H-projective morphism by Morphisms, Lemma 29.43.16. It follows that $Z' \to \mathop{\mathrm{Spec}}(\mathbf{Z})$ is H-projective by Morphisms, Lemma 29.43.7. Thus there exists a closed immersion $Z' \to \mathbf{P}^ m_{\mathop{\mathrm{Spec}}(\mathbf{Z})}$ for some $m \geq 0$. It follows that the diagonal morphism

\[ X' \to Y \times \mathbf{P}^ m_\mathbf {Z} = \mathbf{P}^ m_ Y \]

is an immersion (because the composition with the projection to $\mathbf{P}^ m_\mathbf {Z}$ is an immersion) and we win. $\square$


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 088U. Beware of the difference between the letter 'O' and the digit '0'.