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. $\square$

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