Chow's lemma

Proof

The proof here is a standard one (cf. EGA II, 5.6.1).

It is easy to reduce to the case when X is irreducible, as follows. X is noetherian since it is of finite type over a noetherian base. Then it's also topologically noetherian, and consists of a finite number of irreducible components X i , which are each proper over S (because they're closed immersions in the scheme X which is proper over S ). If, within each of these irreducible components, there exists a dense open U i ⊂ X i , then we can take U := ⨆ ( U i ∖ ⋃ i ≠ j X j ) . It is not hard to see that each of the disjoint pieces are dense in their respective X i , so the full set U is dense in X . In addition, it's clear that we can similarly find a morphism g which satisfies the density condition.

Having reduced the problem, we now assume X is irreducible. We recall that it must also be noetherian. Thus, we can find a finite open affine cover X = ⋃ i = 1 n U i . U i are quasi-projective over S ; there are open immersions over S , ϕ i : U i → P i into some projective S -schemes P i . Put U = ∩ U i . U is nonempty since X is irreducible. Let

be given by ϕ i 's restricted to U over S . Let

be given by U ↪ X and ϕ over S . ψ is then an immersion; thus, it factors as an open immersion followed by a closed immersion X ′ → X × S P . Let f : X ′ → X be the immersion followed by the projection. We claim f induces f − 1 ( U ) ≃ U ; for that, it is enough to show f − 1 ( U ) = ψ ( U ) . But this means that ψ ( U ) is closed in U × S P . ψ factorizes as U → Γ ϕ U × S P → X × S P . P is separated over S and so the graph morphism Γ ϕ is a closed immersion. This proves our contention.

It remains to show X ′ is projective over S . Let g : X ′ → P be the closed immersion followed by the projection. Showing that g is a closed immersion shows X ′ is projective over S . This can be checked locally. Identifying U i with its image in P i we suppress ϕ i from our notation.

Let V i = p i − 1 ( U i ) where p i : P → P i . We claim g − 1 ( V i ) are an open cover of X ′ . This would follow from f − 1 ( U i ) ⊂ g − 1 ( V i ) as sets. This in turn follows from f = p i ∘ g on U i as functions on the underlying topological space. Thus it is enough to show that for each i the map g : g − 1 ( V i ) → V i , denoted by h , is a closed immersion (since the property of being a closed immersion is local on the base).

Fix i . Let Z be the graph of u : V i → p i U i ↪ X . It is a closed subscheme of X × S V i since X is separated over S . Let q 1 : X × S P → X ,   q 2 : X × S P → P be the projections. We claim that h factors through Z , which would imply h is a closed immersion. But for w : U ′ → V i we have:

The last equality holds and thus there is w that satisfies the first equality. This proves our claim. ◻

Additional statements

In the statement of Chow's lemma, if X is reduced, irreducible, or integral, we can assume that the same holds for X ′ . If both X and X ′ are irreducible, then f : X ′ → X is a birational morphism. (cf. EGA II, 5.6).