Chow's lemma

Chow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry. It roughly says that a proper morphism is fairly close to being a projective morphism. More precisely, a version of it states the following:

If [math]\displaystyle{ X }[/math] is a scheme that is proper over a noetherian base [math]\displaystyle{ S }[/math], then there exists a projective [math]\displaystyle{ S }[/math]-scheme [math]\displaystyle{ X' }[/math] and a surjective [math]\displaystyle{ S }[/math]-morphism [math]\displaystyle{ f: X' \to X }[/math] that induces an isomorphism [math]\displaystyle{ f^{-1}(U) \simeq U }[/math] for some dense open [math]\displaystyle{ U\subseteq X. }[/math]

Proof

The proof here is a standard one.

Reduction to the case of [math]\displaystyle{ X }[/math] irreducible

We can first reduce to the case where [math]\displaystyle{ X }[/math] is irreducible. To start, [math]\displaystyle{ X }[/math] is noetherian since it is of finite type over a noetherian base. Therefore it has finitely many irreducible components [math]\displaystyle{ X_i }[/math], and we claim that for each [math]\displaystyle{ X_i }[/math] there is an irreducible proper [math]\displaystyle{ S }[/math]-scheme [math]\displaystyle{ Y_i }[/math] so that [math]\displaystyle{ Y_i\to X }[/math] has set-theoretic image [math]\displaystyle{ X_i }[/math] and is an isomorphism on the open dense subset [math]\displaystyle{ X_i\setminus \cup_{j\neq i} X_j }[/math] of [math]\displaystyle{ X_i }[/math]. To see this, define [math]\displaystyle{ Y_i }[/math] to be the scheme-theoretic image of the open immersion

[math]\displaystyle{ X\setminus \cup_{j\neq i} X_j \to X. }[/math]

Since [math]\displaystyle{ X\setminus \cup_{j\neq i} X_j }[/math] is set-theoretically noetherian for each [math]\displaystyle{ i }[/math], the map [math]\displaystyle{ X\setminus \cup_{j\neq i} X_j\to X }[/math] is quasi-compact and we may compute this scheme-theoretic image affine-locally on [math]\displaystyle{ X }[/math], immediately proving the two claims. If we can produce for each [math]\displaystyle{ Y_i }[/math] a projective [math]\displaystyle{ S }[/math]-scheme [math]\displaystyle{ Y_i' }[/math] as in the statement of the theorem, then we can take [math]\displaystyle{ X' }[/math] to be the disjoint union [math]\displaystyle{ \coprod Y_i' }[/math] and [math]\displaystyle{ f }[/math] to be the composition [math]\displaystyle{ \coprod Y_i' \to \coprod Y_i\to X }[/math]: this map is projective, and an isomorphism over a dense open set of [math]\displaystyle{ X }[/math], while [math]\displaystyle{ \coprod Y_i' }[/math] is a projective [math]\displaystyle{ S }[/math]-scheme since it is a finite union of projective [math]\displaystyle{ S }[/math]-schemes. Since each [math]\displaystyle{ Y_i }[/math] is proper over [math]\displaystyle{ S }[/math], we've completed the reduction to the case [math]\displaystyle{ X }[/math] irreducible.

[math]\displaystyle{ X }[/math] can be covered by finitely many quasi-projective [math]\displaystyle{ S }[/math]-schemes

Next, we will show that [math]\displaystyle{ X }[/math] can be covered by a finite number of open subsets [math]\displaystyle{ U_i }[/math] so that each [math]\displaystyle{ U_i }[/math] is quasi-projective over [math]\displaystyle{ S }[/math]. To do this, we may by quasi-compactness first cover [math]\displaystyle{ S }[/math] by finitely many affine opens [math]\displaystyle{ S_j }[/math], and then cover the preimage of each [math]\displaystyle{ S_j }[/math] in [math]\displaystyle{ X }[/math] by finitely many affine opens [math]\displaystyle{ X_{jk} }[/math] each with a closed immersion in to [math]\displaystyle{ \mathbb{A}^n_{S_j} }[/math] since [math]\displaystyle{ X\to S }[/math] is of finite type and therefore quasi-compact. Composing this map with the open immersions [math]\displaystyle{ \mathbb{A}^n_{S_j}\to \mathbb{P}^n_{S_j} }[/math] and [math]\displaystyle{ \mathbb{P}^n_{S_j} \to \mathbb{P}^n_S }[/math], we see that each [math]\displaystyle{ X_{ij} }[/math] is a closed subscheme of an open subscheme of [math]\displaystyle{ \mathbb{P}^n_S }[/math]. As [math]\displaystyle{ \mathbb{P}^n_S }[/math] is noetherian, every closed subscheme of an open subscheme is also an open subscheme of a closed subscheme, and therefore each [math]\displaystyle{ X_{ij} }[/math] is quasi-projective over [math]\displaystyle{ S }[/math].

Construction of [math]\displaystyle{ X' }[/math] and [math]\displaystyle{ f:X'\to X }[/math]

Now suppose [math]\displaystyle{ \{U_i\} }[/math] is a finite open cover of [math]\displaystyle{ X }[/math] by quasi-projective [math]\displaystyle{ S }[/math]-schemes, with [math]\displaystyle{ \phi_i:U_i\to P_i }[/math] an open immersion in to a projective [math]\displaystyle{ S }[/math]-scheme. Set [math]\displaystyle{ U=\cap_i U_i }[/math], which is nonempty as [math]\displaystyle{ X }[/math] is irreducible. The restrictions of the [math]\displaystyle{ \phi_i }[/math] to [math]\displaystyle{ U }[/math] define a morphism

[math]\displaystyle{ \phi: U \to P = P_1 \times_S \cdots \times_S P_n }[/math]

so that [math]\displaystyle{ U\to U_i\to P_i = U\stackrel{\phi}{\to} P \stackrel{p_i}{\to} P_i }[/math], where [math]\displaystyle{ U\to U_i }[/math] is the canonical injection and [math]\displaystyle{ p_i:P\to P_i }[/math] is the projection. Letting [math]\displaystyle{ j:U\to X }[/math] denote the canonical open immersion, we define [math]\displaystyle{ \psi=(j,\phi)_S: U\to X\times_S P }[/math], which we claim is an immersion. To see this, note that this morphism can be factored as the graph morphism [math]\displaystyle{ U\to U\times_S P }[/math] (which is a closed immersion as [math]\displaystyle{ P\to S }[/math] is separated) followed by the open immersion [math]\displaystyle{ U\times_S P\to X\times_S P }[/math]; as [math]\displaystyle{ X\times_S P }[/math] is noetherian, we can apply the same logic as before to see that we can swap the order of the open and closed immersions.

Now let [math]\displaystyle{ X' }[/math] be the scheme-theoretic image of [math]\displaystyle{ \psi }[/math], and factor [math]\displaystyle{ \psi }[/math] as

[math]\displaystyle{ \psi:U\stackrel{\psi'}{\to} X'\stackrel{h}{\to} X\times_S P }[/math]

where [math]\displaystyle{ \psi' }[/math] is an open immersion and [math]\displaystyle{ h }[/math] is a closed immersion. Let [math]\displaystyle{ q_1:X\times_S P\to X }[/math] and [math]\displaystyle{ q_2:X\times_S P\to P }[/math] be the canonical projections. Set

[math]\displaystyle{ f:X'\stackrel{h}{\to} X\times_S P \stackrel{q_1}{\to} X, }[/math]

[math]\displaystyle{ g:X'\stackrel{h}{\to} X\times_S P \stackrel{q_2}{\to} P. }[/math]

We will show that [math]\displaystyle{ X' }[/math] and [math]\displaystyle{ f }[/math] satisfy the conclusion of the theorem.

Verification of the claimed properties of [math]\displaystyle{ X' }[/math] and [math]\displaystyle{ f }[/math]

To show [math]\displaystyle{ f }[/math] is surjective, we first note that it is proper and therefore closed. As its image contains the dense open set [math]\displaystyle{ U\subset X }[/math], we see that [math]\displaystyle{ f }[/math] must be surjective. It is also straightforward to see that [math]\displaystyle{ f }[/math] induces an isomorphism on [math]\displaystyle{ U }[/math]: we may just combine the facts that [math]\displaystyle{ f^{-1}(U)=h^{-1}(U\times_S P) }[/math] and [math]\displaystyle{ \psi }[/math] is an isomorphism on to its image, as [math]\displaystyle{ \psi }[/math] factors as the composition of a closed immersion followed by an open immersion [math]\displaystyle{ U\to U\times_S P \to X\times_S P }[/math]. It remains to show that [math]\displaystyle{ X' }[/math] is projective over [math]\displaystyle{ S }[/math].

We will do this by showing that [math]\displaystyle{ g:X'\to P }[/math] is an immersion. We define the following four families of open subschemes:

[math]\displaystyle{ V_i = \phi_i(U_i)\subset P_i }[/math]

[math]\displaystyle{ W_i = p_i^{-1}(V_i)\subset P }[/math]

[math]\displaystyle{ U_i' = f^{-1}(U_i)\subset X' }[/math]

[math]\displaystyle{ U_i'' = g^{-1}(W_i)\subset X'. }[/math]

As the [math]\displaystyle{ U_i }[/math] cover [math]\displaystyle{ X }[/math], the [math]\displaystyle{ U_i' }[/math] cover [math]\displaystyle{ X' }[/math], and we wish to show that the [math]\displaystyle{ U_i'' }[/math] also cover [math]\displaystyle{ X' }[/math]. We will do this by showing that [math]\displaystyle{ U_i'\subset U_i'' }[/math] for all [math]\displaystyle{ i }[/math]. It suffices to show that [math]\displaystyle{ p_i\circ g|_{U_i'}:U_i'\to P_i }[/math] is equal to [math]\displaystyle{ \phi_i\circ f|_{U_i'}:U_i'\to P_i }[/math] as a map of topological spaces. Replacing [math]\displaystyle{ U_i' }[/math] by its reduction, which has the same underlying topological space, we have that the two morphisms [math]\displaystyle{ (U_i')_{red}\to P_i }[/math] are both extensions of the underlying map of topological space [math]\displaystyle{ U\to U_i\to P_i }[/math], so by the reduced-to-separated lemma they must be equal as [math]\displaystyle{ U }[/math] is topologically dense in [math]\displaystyle{ U_i }[/math]. Therefore [math]\displaystyle{ U_i'\subset U_i'' }[/math] for all [math]\displaystyle{ i }[/math] and the claim is proven.

The upshot is that the [math]\displaystyle{ W_i }[/math] cover [math]\displaystyle{ g(X') }[/math], and we can check that [math]\displaystyle{ g }[/math] is an immersion by checking that [math]\displaystyle{ g|_{U_i''}:U_i''\to W_i }[/math] is an immersion for all [math]\displaystyle{ i }[/math]. For this, consider the morphism

[math]\displaystyle{ u_i:W_i\stackrel{p_i}{\to} V_i\stackrel{\phi_i^{-1}}{\to} U_i\to X. }[/math]

Since [math]\displaystyle{ X\to S }[/math] is separated, the graph morphism [math]\displaystyle{ \Gamma_{u_i}:W_i\to X\times_S W_i }[/math] is a closed immersion and the graph [math]\displaystyle{ T_i=\Gamma_{u_i}(W_i) }[/math] is a closed subscheme of [math]\displaystyle{ X\times_S W_i }[/math]; if we show that [math]\displaystyle{ U\to X\times_S W_i }[/math] factors through this graph (where we consider [math]\displaystyle{ U\subset X' }[/math] via our observation that [math]\displaystyle{ f }[/math] is an isomorphism over [math]\displaystyle{ f^{-1}(U) }[/math] from earlier), then the map from [math]\displaystyle{ U_i'' }[/math] must also factor through this graph by construction of the scheme-theoretic image. Since the restriction of [math]\displaystyle{ q_2 }[/math] to [math]\displaystyle{ T_i }[/math] is an isomorphism onto [math]\displaystyle{ W_i }[/math], the restriction of [math]\displaystyle{ g }[/math] to [math]\displaystyle{ U_i'' }[/math] will be an immersion into [math]\displaystyle{ W_i }[/math], and our claim will be proven. Let [math]\displaystyle{ v_i }[/math] be the canonical injection [math]\displaystyle{ U\subset X' \to X\times_S W_i }[/math]; we have to show that there is a morphism [math]\displaystyle{ w_i:U\subset X'\to W_i }[/math] so that [math]\displaystyle{ v_i=\Gamma_{u_i}\circ w_i }[/math]. By the definition of the fiber product, it suffices to prove that [math]\displaystyle{ q_1\circ v_i= u_i\circ q_2\circ v_i }[/math], or by identifying [math]\displaystyle{ U\subset X }[/math] and [math]\displaystyle{ U\subset X' }[/math], that [math]\displaystyle{ q_1\circ\psi=u_i\circ q_2\circ \psi }[/math]. But [math]\displaystyle{ q_1\circ\psi = j }[/math] and [math]\displaystyle{ q_2\circ\psi=\phi }[/math], so the desired conclusion follows from the definition of [math]\displaystyle{ \phi:U\to P }[/math] and [math]\displaystyle{ g }[/math] is an immersion. Since [math]\displaystyle{ X'\to S }[/math] is proper, any [math]\displaystyle{ S }[/math]-morphism out of [math]\displaystyle{ X' }[/math] is closed, and thus [math]\displaystyle{ g:X'\to P }[/math] is a closed immersion, so [math]\displaystyle{ X' }[/math] is projective. [math]\displaystyle{ \blacksquare }[/math]

Additional statements

In the statement of Chow's lemma, if [math]\displaystyle{ X }[/math] is reduced, irreducible, or integral, we can assume that the same holds for [math]\displaystyle{ X' }[/math]. If both [math]\displaystyle{ X }[/math] and [math]\displaystyle{ X' }[/math] are irreducible, then [math]\displaystyle{ f: X' \to X }[/math] is a birational morphism.

Bibliography

• Grothendieck, Alexandre; Dieudonné, Jean (1961). "Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes". Publications Mathématiques de l'IHÉS 8. doi:10.1007/bf02699291. http://www.numdam.org/item/PMIHES_1961__8__5_0. 
• Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9