CHOW’S LEMMA

HANSHENG DIAO

1. Projective Morphisms In order to state the Chow’s Lemma, we need the notion of pro- jective morphisms of schemes. Here we adopt the definition used in Hartshorne’s Algebraic Geometry.

n Notation 1. • For any ring A, let PA denote the projective n- space over A; i.e., n PA := Proj A[x0, x1, ..., xn]. n • For any scheme X and any positive integer n, define PX to be the projective n-space over X; i.e., n := n × X. PX PZ Spec Z Definition 2. • A morphism f : X → Y of schemes is said to n be projective if it factors into a closed embedding X → PY for n some n, followed by the projection PY → Y . • A morphism f : X → Y of schemes is said to be quasi-projective if it factors into an open embedding X → X0 followed by a projective morphism X0 → Y . Proposition 3. (1) A closed embedding is projective. (2) A composition of projective morphisms is projective. (3) Projectiveness is stable under base change. (4) A product of projective morphisms are projective. Proposition 4. (1) Open and closed embeddings are quasi-projective. (2) Every quasi-affine morphism of finite type is quasi-projective. (3) A composition of quasi-projective morphisms is quasi-projective. (4) Quasi-projectiveness is stable under base change. (5) A product of quasi-projective morphisms are quasi-projective. Proposition 5. (1) Every projective morphism is quasi-projective and proper. (2) If Y is a quasi-compact scheme, then every proper quasi-projective morphism f : X → Y is projective. 1 2 HANSHENG DIAO

For the proof of Proposition 3, 4, and 5, please refer to Hartshorne II.4 and EGA II.5.4. In the proof of Chow’s Lemma, we need the following notion of scheme-theoretic image of a morphism.

Proposition 6. (1) Let f : X → Y be a morphism of schemes such that f∗(OX ) is a quasi-coherent OY -module. Then there exists a minimal closed subscheme Y 0 of Y such that f factors through the injection j : Y 0 → Y . Moreover, Y 0 is homeomorphic to f(X) as topological subspaces in Y . This Y 0 is called the scheme-theoretic image of f : X → Y in Y . (2) Suppose X is a subscheme of Y and the embedding i : X,→ Y is quasi-compact. Then the scheme-theoretic image of i is just the closure X of X in Y , and the embedding i factors into an open embedding X,→ X followed by a closed embedding X → Y .

For the proof of Proposition 6, please refer to EGA I.9.5.

2. Chow’s Lemma By Proposition 5, every projective morphism is proper. On the other hand, Chow’s lemma tells us that a proper morphism is not too far from a projective one.

Theorem 7 (Chow’s Lemma). Let f : X → S be a proper morphism of schemes and S is noetherian. Then there exists a scheme X0 and a morphism g : X0 → X over S such that X0 is projective over S, and there is an open dense subset U ⊆ X such that g induces an isomorphism g−1(U) →∼ U.

Proof. Step 1: We can reduce to the case X irreducible. Since S is noetherian and f : X → S is proper, X is also noe- therian (because X is locally noetherian and quasi-compact). So we can write X as union of finitely many irreducible components; say Sn X = i=1 Xi. Since Xi are closed in X, the maps fi : Xi → S re- 0 main proper. Suppose we can find Xi and Ui for each map Xi → S 0 Fn 0 satisfying the conditions in Chow’s Lemma. We take X = i=1 Xi Fn S 0 and take U = i=1(Ui\ j6=i Xj). Then X → S is also projective and −1 Fn −1 ∼ Fn g (U) = i=1 gi (Ui) → i=1 Ui = U. CHOW’S LEMMA 3

Step 2: Now assume X is irreducible. Since S is noetherian, we can find a finite affine open covering

d d [ [ S = Si = Spec Ai. i=1 i=1 Since f is of finite type, we can find, for each i, a finite affine open covering mi mi −1 [ [ f (Si) = Ui,j = Spec Bi,j j=1 i=1 such that Bi,j is a finitely generated Ai-algebra. Note that Ui,j → Si are affine and of finite type, thus quasi-projective. We also know that the open embeddings Si ,→ S are quasi-projective. ˜ Hence fi,j = f|Ui,j : Ui,j → S is also quasi-projective.

Step 3: Construction of X0 Relabel {Ui,j} by {V1, ..., Vn}. For each k ∈ {1, 2, ..., n}, we have quasi- projective morphism fk = f|Vk : Vk → S. In particular, fk factors into an open embedding φk : Vk → Pk followed by a projective morphism Pk → S. Tn Put U = k=1 Vk. Since X is irreducible, U must be dense open in X. Let

φ = (φ1, ..., φn)S : U → P = P1 × P2 × · · · × Pn. S S S

Let qk : P → Pk be the natural projection onto the k-th factor. We have a natural map P → S, which is projective because products of projective morphisms is projective. Now define Ψ = (j, φ): U → X ×P S where j : U,→ X is the natural embedding. Note that Ψ : U → X × P factors into the diagonal map 4 : U → S U × U × · · · × U followed by an open embedding U × U × · · · × U,→ S S S S S S X × P1 × · · · × Pn. Clearly, 4 is a closed embedding. So Ψ is an em- S S S bedding. Let X0 = Ψ(U) be the scheme-theoretic image os Ψ in X ×P . S In particular, we have an open embedding ψ0 : U,→ X0 and a closed embedding τ : X0 → X × P . S Step 4: Construction of g : X0 → X. Let p1 : X × P → X and p2 : X × P → P be the natural projections. S S 0 Let g = p1 ◦ τ : X → X. Let p : X × P → S be the composition of p1 S 4 HANSHENG DIAO and f : X → S. Let ρ = f ◦ g : X0 → S. Then we have the following commutative diagram:

ψ0 τ p U > X0 > X × P 1 > X S

ρ f p

> ∨ S <

Step 5: We show that g−1(U) →∼ U. Clearly, p−1(U) = U × P . Let U 0 = g−1(U) = τ −1(U × P ). Note that S S 0 0 g ◦ ψ = p1 ◦ (τ ◦ ψ ) = p1 ◦ Ψ = j : U,→ X. So U = ψ0(U) ⊆ g−1(U) = U 0. We have a natural open embedding 0 0 0 i : U,→ U . The composition of i : U,→ U and τ|U 0 : U → U × P is S simply the graph morphism Γφ : U → U × P , which is a closed embed- S ding because P is separated over S. So the scheme-theoretic image of Γφ is just U. On the other hand, the scheme-theoretic image of Γφ (as a topological space) is the closure of U in U × P , which is just U 0. So S U ' U 0 = g−1(U), as desired.

Step 6: Finally, we show that ρ : X0 → S is projective. p Note that ρ : X0 → S is the composition of η : X0 →τ X × P →2 P and S a projective morphism P → S. So we only need to show that η is a closed embedding. We also notice that p2 : X × P → P is a closed map because X → S S is proper. So η = p2 ◦ τ is closed. It remains to show that η is an embedding. We have the following diagram:

0 ψ 0 τ p1 U > X > X × P > X < Vk S

p2 η φk > ∨ qk < P > Pk CHOW’S LEMMA 5

0 −1 0 0 −1 00 −1 Let Vk = φk(Vk), Wk = qk (Vk), Uk = g (Vk), and Uk = η (Wk). 0 0 Then {Uk} is an open covering of X .

0 00 Claim: Uk ⊆ Uk . Proof of the claim: It suffices to show that the following diagram is commutative:

η|U0 0 k Uk > P

g|U0 qk k ∨ ∨ φk Vk > Pk

0 0 0 0 Note that Uk is an open subscheme of X containing U . Since U is 0 0 0 dense in X , U is also dense in Uk. If we compose the above diagram 0 0 with the embedding U = U → Uk, we obtain the following diagram:

φ U > P

jk qk ∨ ∨ φk Vk > Pk

where jk : U,→ Vk is the natural embedding. This diagram is obviously commutative. So the original diagram is also commutative 0 0 because U is dense in Uk.

00 0 By the claim, {Uk } is also an open covering of X . Note that {Wk} 0 forms an open covering of η(X ). So we only need to show that η| 00 : Uk 00 Uk → Wk is an embedding. Consider the composition

φ−1 qk 0 k uk : Wk → Vk → Vk ,→ X.

Since X is separated over S, the graph morphism Γuk : Wk → X × Wk S is a closed embedding. Namely, Tk = Γuk (Wk) is a closed subscheme of X × Wk. S 6 HANSHENG DIAO

0 Claim: U ⊆ Tk. 0 Proof of the claim: Let vk : U ,→ X × Wk be the natural em- S 0 bedding. We need to find a morphism wk : U → Wk such that vk = Γuk ◦ wk. By the definition of fibre product, this is equivalent 0 0 to show p1 ◦ vk = uk ◦ p2 ◦ vk. Since ψ : U → U is an isomorphism, it suffices to check that p1 ◦ ψ = uk ◦ p2 ◦ ψ. But this is clear because p1 ◦ ψ = j and p2 ◦ ψ = φ.

With a same argument as in Step 5, we know that the scheme- 0 00 theoretic image of U → X × Wk is just Uk . So by the minimality S 00 of scheme-theoretic image, we know that Uk is an open subscheme of Tk. Now, since the restriction of p2 on Tk is an isomorphism on Wk, 00 the restriction of η on Uk is an embedding in Wk. This completes the proof.  References [1] Robin Hartshorne. Algebraic Geometry. [2] David Eisenbud & Joe Harris. The Geometry of Schemes. [3] A. Grothendieck, EGA I, II