This result says that proper morphisms are fairly close to projective morphisms.
Let \(X\) be proper over a noetherian scheme \(S\). Then there is a scheme \(X'\) and a morphism \(g: X' \to X\) such that \(X'\) is projective over \(S\), and there is an open dense subset \(U \subseteq X\) such that \(g\) induces an isomorphism of \(\inverseof{g}(U)\) onto \(U\). Prove this in the following steps.
Reduce to the case \(X\) irreducible.
Show that \(X\) can be covered by finitely many open subsets \(U_i\), \(i = 1, \ldots, n\), each of which is quasi-projective over \(S\). Let \(U_i \injects P_i\) be an open immersion of \(U_i\) into a scheme \(P_i\) which is projective over \(S\).
Let \(U = \bigcap_i U_i\), and consider the map \[ f: U \to X \fiberproduct{S} P_1 \fiberproduct{S} \cdots \fiberproduct{S} P_n \] deduced from the given maps \(U \to X\) and \(U \to P_i\). Let \(X'\) be the closed image subscheme structure on \(\cl_X f(U)\) (see Ex. 3.11d). Let \(g: X' \to X\) be the projection onto the first factor, and let \(h: X' \to P = P_1 \fiberproduct{S} \cdots \fiberproduct{S} P_n\) be the projection onto the product of the remaining factors. Show that \(h\) is a closed immersion, hence \(X'\) is projective over \(S\).
Show that \(\inverseof{g}(U) \to U\) is an isomorphism, completing the proof.
Solution
1. It is enough to prove the theorem when the underlying topological space of \(X\) is irreducible.
Because \(X\) is proper over the noetherian scheme \(S\), it is noetherian. Let \[ X_1,\ldots,X_m \] be its irreducible components.
For each \(i\), put \[ V_i = X_i\setminus\bigcup_{j\ne i}X_j. \] This is a nonempty open subset of \(X\): it is the part of the component \(X_i\) which lies on no other component.
Let \[ Y_i\hookrightarrow X \] be the scheme-theoretic closure of the open immersion \[ V_i\hookrightarrow X. \] Then \(|Y_i|=X_i\), so \(Y_i\) is irreducible as a topological space. It is a closed subscheme of the proper \(S\)-scheme \(X\), hence is proper over \(S\). Moreover \[ Y_i|_{V_i}=V_i \] scheme-theoretically: over the open set \(V_i\) the given immersion is already a closed immersion onto all of \(V_i\), so its scheme-theoretic image there is \(V_i\) itself.
Assume the theorem known for schemes with irreducible underlying space. For each \(Y_i\) choose a projective \(S\)-scheme \[ Y_i' \] and a morphism \[ g_i:Y_i'\to Y_i \] which is an isomorphism over some dense open subset of \(Y_i\). Intersect that dense open with \(V_i\); after replacing it by this smaller nonempty dense open, write it as \[ W_i\subseteq V_i. \] Then \(g_i\) is an isomorphism over \(W_i\), and the composite \[ Y_i'\xrightarrow{g_i}Y_i\hookrightarrow X \] is an isomorphism over \(W_i\subseteq X\).
Set \[ X'=\coprod_{i=1}^mY_i'. \] A finite disjoint union of projective \(S\)-schemes is projective over \(S\): if \[ Y_i'\hookrightarrow\mathbb P^{n_i}_S, \] embed the finitely many projective spaces into pairwise disjoint coordinate linear subspaces of one sufficiently large projective space over \(S\); their finite union is closed.
The morphisms \(Y_i'\to X\) combine to \[ g:X'\to X. \] The opens \(W_i\) are pairwise disjoint by construction, and \[ W=\bigcup_iW_i \] is open and dense in \(X\): it meets every irreducible component in a dense open subset. Over \(W\) the morphism \(g\) is the disjoint union of the isomorphisms \[ g_i^{-1}(W_i)\xrightarrow{\sim}W_i. \] Hence \(g^{-1}(W)\to W\) is an isomorphism.
Thus the irreducible case implies the general case, including the nonreduced scheme structure on the dense open.
2. We now assume \(X\) irreducible.Every point \(x\in X\) has an open neighborhood which is quasi-projective over \(S\).
Let \(s\in S\) be the image of \(x\). Choose an affine open neighborhood \[ V=\Spec B\subseteq S \] of \(s\). The inverse image \[ X_V=X\times_SV \] is of finite type over \(V\).
Choose an affine open neighborhood \[ W=\Spec A\subseteq X_V \] of \(x\). By Hartshorne II.3.3(c), \(A\) is a finitely generated \(B\)-algebra. Hence there is a surjection \[ B[t_1,\ldots,t_N]\twoheadrightarrow A, \] which gives a closed immersion \[ W\hookrightarrow\mathbb A^N_V. \]
The affine space \(\mathbb A^N_V\) is the standard open subset of \[ \mathbb P^N_V, \] and \(\mathbb P^N_V\) is itself an open subscheme of \[ \mathbb P^N_S \] because \(V\subseteq S\) is open. Thus the composite \[ W\longrightarrow\mathbb P^N_S \] is a locally closed immersion.
Let \(P_W\) be its scheme-theoretic closure in \(\mathbb P^N_S\). Then \[ W\hookrightarrow P_W \] is an open immersion and \[ P_W\hookrightarrow\mathbb P^N_S \] is a closed immersion. Hence \(P_W\) is projective over \(S\), and \(W\) is quasi-projective over \(S\).
3. There is a finite open cover\[ X=U_1\cup\cdots\cup U_n \]such that every \(U_i\) is quasi-projective over \(S\).
The neighborhoods from 2 cover \(X\). Since \(X\) is proper over \(S\), it is of finite type, hence quasi-compact because \(S\) is noetherian and the structure morphism is proper. Thus finitely many of the quasi-projective neighborhoods suffice.
4. For each \(i\), choose an open immersion\[ j_i:U_i\hookrightarrow P_i \]with \(P_i\) projective over \(S\).The open subset\[ U=\bigcap_{i=1}^nU_i \]is nonempty and dense in \(X\).
The first assertion is the definition of quasi-projectivity.
Since \(X\) is irreducible, every nonempty open subset is dense, and any finite intersection of nonempty open subsets is nonempty. Thus the finite intersection \(U\) is nonempty, open, and dense.
5. Put\[ P=P_1\times_S\cdots\times_SP_n. \]Then \(P\) is projective over \(S\).
Projective morphisms are stable under finite products by Hartshorne II.4.8 together with II.4.9. Hence the structure morphism \[ P\to S \] is projective.
6. The maps\[ U\hookrightarrow X, \qquad U\hookrightarrow U_i\xrightarrow{j_i}P_i \]define a morphism\[ f:U\longrightarrow Q:=X\times_SP, \]given on points by\[ x\longmapsto\bigl(x,j_1(x),\ldots,j_n(x)\bigr). \]Let\[ X'\hookrightarrow Q \]be the scheme-theoretic image of \(f\).
The universal property of the fibre product gives the displayed morphism from its coordinate maps.
Since \(U\) is noetherian, the morphism \(f\) is quasi-compact and quasi-separated. Hartshorne II.3.11 therefore supplies its scheme-theoretic image, a closed subscheme \(X'\subseteq Q\) through which \(f\) factors minimally.
7. Let\[ g:X'\to X, \qquad h:X'\to P \]be the restrictions of the two projections from \(Q=X\times_SP\).The morphism \(h\) is proper.
The projection \[ Q=X\times_SP\longrightarrow P \] is the base change of the proper morphism \[ X\longrightarrow S \] along \(P\to S\). Hence \(Q\to P\) is proper.
The inclusion \(X'\hookrightarrow Q\) is a closed immersion, hence proper. Therefore their composite \[ h:X'\to P \] is proper.
8. For each \(i\), let\[ O_i=U_i\times_SP\subseteq Q. \]Then the opens \(O_i\) cover \(Q\), and hence\[ X_i'=X'\cap O_i \]cover \(X'\).
The opens \(U_i\) cover \(X\), so their inverse images under the projection \[ Q\to X \] cover \(Q\). Intersecting with the closed subscheme \(X'\) gives an open cover of \(X'\).
9. Let\[ \pi_i:P\to P_i \]be the \(i\)th projection and define\[ G_i = U_i\times_{P_i}P, \]where \(U_i\to P_i\) is \(j_i\) and \(P\to P_i\) is \(\pi_i\).Then\[ G_i\to P \]is an open immersion.
The morphism \(G_i\to P\) is the base change of the open immersion \[ j_i:U_i\hookrightarrow P_i \] along \(\pi_i:P\to P_i\). Open immersions are stable under base change.
10. There is a closed immersion\[ G_i\hookrightarrow O_i=U_i\times_SP. \]
The morphism is the graph of the composite \[ U_i\xrightarrow{j_i}P_i \] in the \(i\)th projective factor, together with the unrestricted remaining factors.
More explicitly, the graph \[ \Gamma_{j_i}:U_i\longrightarrow U_i\times_SP_i \] is a closed immersion because \(P_i\) is separated over \(S\); projective morphisms are separated. Base changing this graph along the product of the remaining factors gives \[ G_i\hookrightarrow U_i\times_SP=O_i. \] Thus it is a closed immersion.
11. The image \(f(U)\) is contained in every \(G_i\subseteq O_i\), and consequently\[ X_i'=X'\cap O_i \]is a closed subscheme of \(G_i\).
For \(x\in U\), the \(i\)th projective coordinate of \(f(x)\) is by definition \(j_i(x)\). Hence \[ f(U)\subseteq G_i. \]
10. The restriction of the scheme-theoretic image \(X'\) to \(O_i\) is the scheme-theoretic image of\[ f^{-1}(O_i)=U\longrightarrow O_i. \]Because this morphism factors through the closed subscheme \(G_i\), minimality of the scheme-theoretic image gives a factorization\[ X_i'\hookrightarrow G_i. \]This factor is again a closed immersion.
12. The restriction\[ h|_{X_i'}:X_i'\longrightarrow P \]is an immersion for every \(i\).
By 11, \[ X_i'\hookrightarrow G_i \] is a closed immersion, and by 9, \[ G_i\hookrightarrow P \] is an open immersion. Their composition is therefore an immersion.
13. The morphism\[ h:X'\to P \]is an immersion.
12. Hence \(h\) is an immersion.
14. A proper immersion is a closed immersion.Hence\[ \boxed{h:X'\hookrightarrow P} \]is a closed immersion.
An immersion factors as an open immersion followed by a closed immersion onto a locally closed subscheme of the target. A proper morphism is closed, so the image of the proper immersion \(h\) is closed in \(P\). Thus the locally closed image is actually closed, and the immersion is a closed immersion.
Equivalently, a proper monomorphism is a closed immersion.
15. The scheme \(X'\) is projective over \(S\).
By 5, \(P\) is projective over \(S\). By 14, \(X'\) is a closed subscheme of \(P\). A closed immersion followed by a projective morphism is projective. Hence \[ X'\to S \] is projective.
16. Inside the open subscheme\[ O=U\times_SP\subseteq Q, \]the morphism\[ f:U\to O \]is a closed immersion.
The morphism \(U\to P\) is the product of the maps \[ j_i|_U:U\to P_i. \] The scheme \(P\) is separated over \(S\), being projective. Therefore the graph \[ U\longrightarrow U\times_SP=O \] of \(U\to P\) is a closed immersion. This graph is precisely \(f\).
17. The restriction of \(X'\) to \(O\) is exactly \(f(U)\) scheme-theoretically:\[ \boxed{X'\cap O=f(U).} \]
The scheme-theoretic image commutes with restriction to an open subscheme for the quasi-compact morphism \(f\). Thus \[ X'\cap O \] is the scheme-theoretic image of \[ f:U\to O. \] But 16 shows that this map is already a closed immersion. Its scheme-theoretic image is therefore its closed image \(f(U)\) itself.
18. The inverse image of \(U\subseteq X\) under \(g\) is\[ g^{-1}(U)=X'\cap O=f(U). \]
The inverse image under the projection \[ g:X'\to X \] of \(U\) is the intersection of \(X'\) with the inverse image of \(U\) in \(Q=X\times_SP\), namely \[ U\times_SP=O. \] Apply 17.
19. The morphism\[ \boxed{g^{-1}(U)\xrightarrow{\sim}U} \]is an isomorphism.
By 18, the source is the graph \(f(U)\). The first projection \[ f(U)\longrightarrow U \] is inverse to the graph morphism \[ f:U\longrightarrow f(U). \] Thus \(g\) restricts to an isomorphism over \(U\).
20. This proves Chow’s lemma.
In the irreducible case, <1>3–<1>4 produce the finite quasi-projective cover and dense open \(U\) requested in part (b). Steps <1>5–<1>15 construct the projective scheme \(X'\) and prove part (c). Steps <1>16–<1>19 prove part (d).
Finally 1 reduces the general noetherian proper scheme to this irreducible case. Hence there exist a projective \(S\)-scheme \(X'\) and a morphism \[ g:X'\to X \] which is an isomorphism over a dense open subset of \(X\).
21. Q.E.D.
Step 20 is the stated theorem.