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
\[ \xymatrix{ X \ar[rd] & X' \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^ n_ S \ar[dl] \\ & S & } \]
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 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
\[ \xymatrix{ \eta _ j \in Z_ j \ar[r] & X \ar[r] \ar[rd] & Y \ar[r] \ar[d] & Y_ i \ar[d] \\ & & S \ar[r] & S_ i } \]
Denote $h : X \to Y_ i$ the composition.
For $i' \geq i$ write $Y_{i'} = S_{i'} \times _{S_ i} Y_ i$. Then $Y = \mathop{\mathrm{lim}}\nolimits _{i' \geq i} Y_{i'}$, 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
\[ \xymatrix{ Y' \ar[rd] & Y^* \ar[d] \ar[l]^\pi \ar[r] & \mathbf{P}^ n_{S_ i} \ar[dl] \\ & S_ i & } \]
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.
$\square$
Comments (1)
Comment #2 by Johan on