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## Tag: 0203

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Lemma 28.10.2. 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 $$\xymatrix{ X \ar[rd] & X' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_S \ar[dl] \\ & S & }$$ where $X' \to \mathbf{P}^n_S$ is an immersion, and $\pi : X' \to X$ is proper and surjective. Moreover, there exists an open dense subscheme $U \subset X$ such that $\pi^{-1}(U) \to U$ is an isomorphism of schemes.

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 26.16.1 using the general Properties, Lemma 24.27.4 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 28.10.1 above. This is what we will do here.

By Proposition 28.7.4 we can find a closed immersion $X \to Y$ where $Y$ is separated and of finite presentation over $S$. Write $S = \mathop{\rm lim}\nolimits_i S_i$ as a directed limit of Noetherian schemes, see Proposition 28.4.4. By Lemma 28.8.1 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 28.6.5 we may assume that $Y_i \to S_i$ is separated. We have the following diagram $$\xymatrix{ \eta_j \in Z_j \ar[r] & X \ar[r] \ar[rd] & Y \ar[r] \ar[d] & Y_i \ar[d] \\ & & S \ar[r] & S_i }$$ Denote $h : X \to Y_i$ the composition.

For $i' \geq i$ write $Y_{i'} = S_{i'} \times_{S_i} Y_i$. Then $Y = \mathop{\rm lim}\nolimits_{i' \geq i} Y_{i'}$, see Lemma 28.2.3. Choose $j, j' \in \{1, \ldots, n\}$, $j \not = j'$. Note that $\eta_j$ is not a specialization of $\eta_{j'}$. By Lemma 28.3.2 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 25.6.2. 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 25.6.3 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 25.8.3. 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 26.16.1 above to the morphism $Y' \to S_i$. This gives a diagram $$\xymatrix{ Y' \ar[rd] & Y^* \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_{S_i} \ar[dl] \\ & S_i & }$$ 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 $$\xymatrix{ X' \ar@{=}[r] & X \times_{Y'} Y^* \ar[r] \ar[d] & Y^* \ar[r] \ar[d] & \mathbf{P}^n_{S_i} \ar[ddl] \\ & X \ar[r] \ar[d] & Y' \ar[d] & \\ & S \ar[r] & S_i & }$$ 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. $\square$

\begin{lemma}
\label{lemma-chow-EGA}
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
$$\xymatrix{ X \ar[rd] & X' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_S \ar[dl] \\ & S & }$$
where $X' \to \mathbf{P}^n_S$ is an immersion, and
$\pi : X' \to X$ is proper and surjective. Moreover, there exists
an open dense subscheme $U \subset X$ such that $\pi^{-1}(U) \to U$
is an isomorphism of schemes.
\end{lemma}

\begin{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.

\medskip\noindent
There are (at least) two ways to proceed with the proof.
The first is to redo the proof of
Cohomology of Schemes, Lemma \ref{coherent-lemma-chow-Noetherian}
using the general
Properties, Lemma \ref{properties-lemma-point-and-maximal-points-affine}
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 \ref{lemma-chow-finite-type} above.
This is what we will do here.

\medskip\noindent
By Proposition \ref{proposition-separated-closed-in-finite-presentation}
we can find a closed immersion $X \to Y$ where $Y$ is separated
and of finite presentation over $S$.
Write $S = \lim_i S_i$ as a directed limit of Noetherian schemes, see
Proposition \ref{proposition-approximate}. By
Lemma \ref{lemma-descend-finite-presentation} 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 \ref{lemma-descend-separated-finite-presentation}
we may assume that $Y_i \to S_i$ is separated.
We have the following diagram
$$\xymatrix{ \eta_j \in Z_j \ar[r] & X \ar[r] \ar[rd] & Y \ar[r] \ar[d] & Y_i \ar[d] \\ & & S \ar[r] & S_i }$$
Denote $h : X \to Y_i$ the composition.

\medskip\noindent
For $i' \geq i$ write $Y_{i'} = S_{i'} \times_{S_i} Y_i$.
Then $Y = \lim_{i' \geq i} Y_{i'}$, see
Lemma \ref{lemma-scheme-over-limit}.
Choose $j, j' \in \{1, \ldots, n\}$, $j \not = j'$.
Note that $\eta_j$ is not a specialization of $\eta_{j'}$.
By Lemma \ref{lemma-topology-limit}
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 \ref{morphisms-definition-scheme-theoretic-image}.
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 \ref{morphisms-lemma-quasi-compact-scheme-theoretic-image}
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 \ref{morphisms-lemma-quasi-compact-dominant}.
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 $Y'$.

\medskip\noindent
We apply Cohomology of Schemes, Lemma
\ref{coherent-lemma-chow-Noetherian} above to the morphism
$Y' \to S_i$. This gives a diagram
$$\xymatrix{ Y' \ar[rd] & Y^* \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^n_{S_i} \ar[dl] \\ & S_i & }$$
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
$$\xymatrix{ X' \ar@{=}[r] & X \times_{Y'} Y^* \ar[r] \ar[d] & Y^* \ar[r] \ar[d] & \mathbf{P}^n_{S_i} \ar[ddl] \\ & X \ar[r] \ar[d] & Y' \ar[d] & \\ & S \ar[r] & S_i & }$$
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.
\end{proof}


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Comment #2 by Johan on July 20, 2012 at 5:39 pm UTC

This version of Chow's lemma is EGA, II, Theorem 5.6.1 b) except that EGA assumes that $S$ is separated. The Noetherian version is 0200.

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