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.<\/p>\n
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.<\/p>\n
No doubt Raynaud and Gruson have in mind a definition a la EGA: we say Z —> W is quasi-projective<\/em> 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. <\/p>\n 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.<\/p>\n 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<\/a> 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).<\/p>\n 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.<\/p>\n 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.<\/p>\n 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.<\/p>\n Surely, you’re not still reading this are you?<\/p>\n","protected":false},"excerpt":{"rendered":" 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 … Continue reading