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30.18 Chow's Lemma

In this section we prove Chow's lemma in the Noetherian case (Lemma 30.18.1). In Limits, Section 32.12 we prove some variants for the non-Noetherian case.

Lemma 30.18.1.reference Let S be a Noetherian scheme. Let f : X \to S be a separated morphism of finite type. Then there exist 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, we may arrange it such that there exists a dense open subscheme U \subset X such that \pi ^{-1}(U) \to U is an isomorphism.

Proof. All of the schemes we will encounter during the rest of the proof are going to be of finite type over the Noetherian scheme S and hence Noetherian (see Morphisms, Lemma 29.15.6). All morphisms between them will automatically be quasi-compact, locally of finite type and quasi-separated, see Morphisms, Lemma 29.15.8 and Properties, Lemmas 28.5.4 and 28.5.8.

The scheme X has only finitely many irreducible components (Properties, Lemma 28.5.7). Say X = X_1 \cup \ldots \cup X_ r is the decomposition of X into irreducible components. Let \eta _ i \in X_ i be the generic point. For every point x \in X there exists an affine open U_ x \subset X which contains x and each of the generic points \eta _ i. See Properties, Lemma 28.29.4. Since X is quasi-compact, we can find a finite affine open covering X = U_1 \cup \ldots \cup U_ m such that each U_ i contains \eta _1, \ldots , \eta _ r. In particular we conclude that the open U = U_1 \cap \ldots \cap U_ m \subset X is a dense open. This and the fact that the U_ i are affine opens covering X are all that we will use below.

Let X^* \subset X be the scheme theoretic closure of U \to X, see Morphisms, Definition 29.6.2. Let U_ i^* = X^* \cap U_ i. Note that U_ i^* is a closed subscheme of U_ i. Hence U_ i^* is affine. Since U is dense in X the morphism X^* \to X is a surjective closed immersion. It is an isomorphism over U. Hence we may replace X by X^* and U_ i by U_ i^* and assume that U is scheme theoretically dense in X, see Morphisms, Definition 29.7.1.

By Morphisms, Lemma 29.39.3 we can find an immersion j_ i : U_ i \to \mathbf{P}_ S^{n_ i} for each i. By Morphisms, Lemma 29.7.7 we can find closed subschemes Z_ i \subset \mathbf{P}_ S^{n_ i} such that j_ i : U_ i \to Z_ i is a scheme theoretically dense open immersion. Note that Z_ i \to S is proper, see Morphisms, Lemma 29.43.5. Consider the morphism

j = (j_1|_ U, \ldots , j_ m|_ U) : U \longrightarrow \mathbf{P}_ S^{n_1} \times _ S \ldots \times _ S \mathbf{P}_ S^{n_ m}.

By the lemma cited above we can find a closed subscheme Z of \mathbf{P}_ S^{n_1} \times _ S \ldots \times _ S \mathbf{P}_ S^{n_ m} such that j : U \to Z is an open immersion and such that U is scheme theoretically dense in Z. The morphism Z \to S is proper. Consider the ith projection

\text{pr}_ i|_ Z : Z \longrightarrow \mathbf{P}^{n_ i}_ S.

This morphism factors through Z_ i (see Morphisms, Lemma 29.6.6). Denote p_ i : Z \to Z_ i the induced morphism. This is a proper morphism, see Morphisms, Lemma 29.41.7 for example. At this point we have that U \subset U_ i \subset Z_ i are scheme theoretically dense open immersions. Moreover, we can think of Z as the scheme theoretic image of the “diagonal” morphism U \to Z_1 \times _ S \ldots \times _ S Z_ m.

Set V_ i = p_ i^{-1}(U_ i). Note that p_ i|_{V_ i} : V_ i \to U_ i is proper. Set X' = V_1 \cup \ldots \cup V_ m. By construction X' has an immersion into the scheme \mathbf{P}^{n_1}_ S \times _ S \ldots \times _ S \mathbf{P}^{n_ m}_ S. Thus by the Segre embedding (see Constructions, Lemma 27.13.6) we see that X' has an immersion into a projective space over S.

We claim that the morphisms p_ i|_{V_ i}: V_ i \to U_ i glue to a morphism X' \to X. Namely, it is clear that p_ i|_ U is the identity map from U to U. Since U \subset X' is scheme theoretically dense by construction, it is also scheme theoretically dense in the open subscheme V_ i \cap V_ j. Thus we see that p_ i|_{V_ i \cap V_ j} = p_ j|_{V_ i \cap V_ j} as morphisms into the separated S-scheme X, see Morphisms, Lemma 29.7.10. We denote the resulting morphism \pi : X' \to X.

We claim that \pi ^{-1}(U_ i) = V_ i. Since \pi |_{V_ i} = p_ i|_{V_ i} it follows that V_ i \subset \pi ^{-1}(U_ i). Consider the diagram

\xymatrix{ V_ i \ar[r] \ar[rd]_{p_ i|_{V_ i}} & \pi ^{-1}(U_ i) \ar[d] \\ & U_ i }

Since V_ i \to U_ i is proper we see that the image of the horizontal arrow is closed, see Morphisms, Lemma 29.41.7. Since V_ i \subset \pi ^{-1}(U_ i) is scheme theoretically dense (as it contains U) we conclude that V_ i = \pi ^{-1}(U_ i) as claimed.

This shows that \pi ^{-1}(U_ i) \to U_ i is identified with the proper morphism p_ i|_{V_ i} : V_ i \to U_ i. Hence we see that X has a finite affine covering X = \bigcup U_ i such that the restriction of \pi is proper on each member of the covering. Thus by Morphisms, Lemma 29.41.3 we see that \pi is proper.

Finally we have to show that \pi ^{-1}(U) = U. To see this we argue in the same way as above using the diagram

\xymatrix{ U \ar[r] \ar[rd] & \pi ^{-1}(U) \ar[d] \\ & U }

and using that \text{id}_ U : U \to U is proper and that U is scheme theoretically dense in \pi ^{-1}(U). \square

Remark 30.18.2. In the situation of Chow's Lemma 30.18.1:

  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).

  2. We may assume that \pi ^{-1}(U) is scheme theoretically dense in X'. Namely, we can simply replace X' by the scheme theoretic closure of \pi ^{-1}(U). In this case we can think of U as a scheme theoretically dense open subscheme of X'. See Morphisms, Section 29.6.

  3. If X is reduced then we may choose X' reduced. This is clear from (2).


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