In this section we prove a number of variants of Chow's lemma. The most interesting version is probably just the Noetherian case, which we stated and proved in Cohomology of Schemes, Section 30.18.
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 where $X' \to \mathbf{P}^ n_ S$ is an immersion, and $\pi : X' \to X$ is proper and surjective.
\[ \xymatrix{ X \ar[rd] & X' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^ n_ S \ar[dl] \\ & S & } \]
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. $\square$
Remark 32.12.2. In the situation of Chow's Lemma 32.12.1:
The morphism $\pi $ is actually H-projective (hence projective, see Morphisms, Lemma 29.43.3) since the morphism $X' \to \mathbf{P}^ n_ S \times _ S X = \mathbf{P}^ n_ X$ is a closed immersion (use the fact that $\pi $ is proper, see Morphisms, Lemma 29.41.7).
We may assume that $X'$ is reduced as we can replace $X'$ by its reduction without changing the other assertions of the lemma.
We may assume that $X' \to X$ is of finite presentation without changing the other assertions of the lemma. This can be deduced from the proof of Lemma 32.12.1 but we can also prove this directly as follows. By (1) we have a closed immersion $X' \to \mathbf{P}^ n_ X$. By Lemma 32.9.4 we can write $X' = \mathop{\mathrm{lim}}\nolimits X'_ i$ where $X'_ i \to \mathbf{P}^ n_ X$ is a closed immersion of finite presentation. In particular $X'_ i \to X$ is of finite presentation, proper, and surjective. For large enough $i$ the morphism $X'_ i \to \mathbf{P}^ n_ S$ is an immersion by Lemma 32.4.16. Replacing $X'$ by $X'_ i$ we get what we want.
Of course in general we can't simultaneously achieve both (2) and (3).
Here is a variant of Chow's lemma where we assume the scheme on top has finitely many irreducible components.
Lemma 32.12.3. Let $S$ be a quasi-compact and quasi-separated scheme. Let $f : X \to S$ be a separated morphism of finite type. Assume that $X$ has finitely many irreducible components. Then there exists an $n \geq 0$ and a diagram where $X' \to \mathbf{P}^ n_ S$ is an immersion, and $\pi : X' \to X$ is proper and surjective. Moreover, there exists an open dense subscheme $U \subset X$ such that $\pi ^{-1}(U) \to U$ is an isomorphism of schemes.
\[ \xymatrix{ X \ar[rd] & X' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^ n_ S \ar[dl] \\ & S & } \]
Proof. Let $X = Z_1 \cup \ldots \cup Z_ n$ be the decomposition of $X$ into irreducible components. Let $\eta _ j \in Z_ j$ be the generic point.
There are (at least) two ways to proceed with the proof. The first is to redo the proof of Cohomology of Schemes, Lemma 30.18.1 using the general Properties, Lemma 28.29.4 to find suitable affine opens in $X$. (This is the “standard” proof.) The second is to use absolute Noetherian approximation as in the proof of Lemma 32.12.1 above. This is what we will do here.
By Proposition 32.9.6 we can find a closed immersion $X \to Y$ where $Y$ is separated and 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 $Y_ i \to S_ i$ of finite presentation so that $Y = S \times _{S_ i} Y_ i$. By Lemma 32.8.6 we may assume that $Y_ i \to S_ i$ is separated. We have the following diagram
Denote $h : X \to Y_ i$ the composition.
For $i' \geq i$ write $Y_{i'} = S_{i'} \times _{S_ i} Y_ i$. Then $Y = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} Y_{i'}$, see Lemma 32.2.3. Choose $j, j' \in \{ 1, \ldots , n\} $, $j \not= j'$. Note that $\eta _ j$ is not a specialization of $\eta _{j'}$. By Lemma 32.4.6 we can replace $i$ by a bigger index and assume that $h(\eta _ j)$ is not a specialization of $h(\eta _{j'})$ for all pairs $(j, j')$ as above. For such an index, let $Y' \subset Y_ i$ be the scheme theoretic image of $h : X \to Y_ i$, see Morphisms, Definition 29.6.2. The morphism $h$ is quasi-compact as the composition of the quasi-compact morphisms $X \to Y$ and $Y \to Y_ i$ (which is affine). Hence by Morphisms, Lemma 29.6.3 the morphism $X \to Y'$ is dominant. Thus the generic points of $Y'$ are all contained in the set $\{ h(\eta _1), \ldots , h(\eta _ n)\} $, see Morphisms, Lemma 29.8.3. Since none of the $h(\eta _ j)$ is the specialization of another we see that the points $h(\eta _1), \ldots , h(\eta _ n)$ are pairwise distinct and are each a generic point of $Y'$.
We apply Cohomology of Schemes, Lemma 30.18.1 above to the morphism $Y' \to S_ i$. This gives a diagram
such that $\pi $ is proper and surjective and an isomorphism over a dense open subscheme $V \subset Y'$. By our choice of $i$ above we know that $h(\eta _1), \ldots , h(\eta _ n) \in V$. Consider the commutative diagram
Note that $X' \to X$ is an isomorphism over the open subscheme $U = h^{-1}(V)$ which contains each of the $\eta _ j$ and hence is dense in $X$. We conclude $X \leftarrow X' \rightarrow \mathbf{P}^ n_ S$ is a solution to the problem posed in the lemma. $\square$
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