# Chow’s lemma

One version of Chow’s lemma is that given a finite type, separated morphism of Noetherian schemes X —> Y, there exists a blowing up X’ —> X with nowhere dense center such that X’ —> Y is quasi-projective.

Chow’s lemma also holds if you replace “schemes” with “algebraic spaces”; see Corollary 5.7.13 of the paper by Raynaud and Gruson. To parse this you have to know what it means for a morphism Z —> W of algebraic spaces to be quasi-projective.

No doubt Raynaud and Gruson have in mind a definition a la EGA: we say Z —> W is quasi-projective if it is representable, of finite type, and there exists an invertible sheaf L on Z such that for every S —> W, where S is an affine scheme, the pullback of L to the fibre product S x_W Z (this is a scheme) is an ample invertible sheaf.

I will show by a very simple example that you cannot use Knutson’s definition and expect Chow’s lemma to hold: Let’s say a morphism of algebraic spaces Z —> W is Knutson-quasi-projective if there exists a factorization Z —> P^n_W —> W where the first arrow is an immersion.

The example is the morphism X = A^1 —> Y = A^1/R where R = Δ ∐ {(t, -t) | t not zero}. In this case Chow’s lemma as formulated above just states that X —> Y is quasi-projective. On the other hand, my faithful readers will remember that in this post we showed that there cannot be an immersion X —> A^n_Y. The exact same argument shows there cannot be an immersion into P^n_Y (or you can easily show that if you have an immersion into P^n_Y, then you also have one into A^n_Y perhaps after a Zariski localization on Y).

The morphism X —> Y above can be “compactified” by embedding X = A^1 into the affine with 0 doubled which is finite etale over Y. So you can find an open immersion of X into an algebraic space finite over Y (this is a general property of quasi-finite separated morphisms). You just cannot find an immersion into the product of P^n and Y.

In the stacks project we don’t yet have defined the notions: relatively ample invertible sheaf, relatively very ample invertible sheaf, quasi-projective morphism, projective morphism for morphisms of algebraic spaces. I think a weaker version of Chow’s lemma that avoids introducing these notions, and is still is somewhat useful, is the following: given a finite type, separated morphism X —> Y with Y Noetherian (say) there exists a blowing up X’ —> X with nowhere dense center and an open immersion of X’ into an algebraic space representable and proper over Y. If Y is a scheme (which is the most important case in applications) you can then use Chow’s lemma for schemes to bootstrap to the statement above.

Knutson proves a version of Chow’s lemma with X’ —> Y Knutson-quasi-projective and with X’ –> X Knutson-projective and birational when both X and Y are separated. As mentioned in the other blog post, I think the problem pointed out above cannot happen if the base algebraic space Y is locally separated. Thus I think it may be possible to generalize Knutson’s version of Chow’s lemma to the case where Y is locally separated.

Surely, you’re not still reading this are you?

## 5 thoughts on “Chow’s lemma”

1. Brian Conrad on said:

In EGA, the concept is ampleness for a line bundle on a scheme (over an affine base, or “over Spec(Z)”) is *only* defined subject to the requirement that “the pre-scheme is a scheme”. That is, it is only defined under a separatedness hypothesis (implicit in the use of the word “schema” in EGA II, 4.5.3, back in the day when the phrase “pre-scheme” was in use). But your example violates this, since R –> A^2 is not a closed immersion. So one can reasonably argue that these examples are not entirely satisfying.

[If one digs into the later EGA errata, according to which EGA II 4.5.2 is applied to any qcqs prescheme X, and likewise for the definition in EGA II 4.5.3, then we should note that 4.5.2(b) forces X to be separated. It seems reasonable to demand that any notion of quasi-projectivity for morphisms should require separatedness, or at least that any counterexample to such a candidate definition should involve a separated morphism in order to qualify as a “good” counterexample.]

2. Brian Conrad on said:

Also, if you look at the discussion of relative ampleness in EGA 4.6.1ff, one sees that they note almost immediately that their definition forces separatedness of the structure morphism.

• Yes, I agree with this and your previous comment. And the morphism X —> Y in my post is separated… and as you say if Z —> W is a quasi-projective morphism of algebraic spaces (as defined in the post), then Z —> W is separated.

All I wanted to do in the post is point out that in order to use the result in Raynaud-Gruson you cannot use the Knutson definition of a quasi-projective morphism but you *have* to use the EGA definition. And I do think the example shows that.

• Brian Conrad on said:

Whoops, I got confused due to the fact that X x_Y X = R isn’t closed in X x X, which is to say that Y isn’t separated, an irrelevant property. OK, so my comments above are largely beside the point. Well, I guess it’s good to be “reminded” that the Chow’s Lemma in Knutson’s book has the separatedness hypothesis on the base, which I had forgotten (and which is fortunately not a severe restriction on the ways that Chow’s Lemma is used in various important applications).

• Yes, as far as I know, the most used case is just where the base is the spectrum of a Noetherian ring.