Grothendieck existence again

Let me try again to find a tweak to the proof of Grothendieck’s existence theorem.

Let X be a Noetherian scheme, Z ⊂ X a closed subscheme, and U ⊂ X the complement. Denote X’ the completion of X along Z (this is a formal scheme). Suppose that we have a triple (F, G, a) where F is a coherent O_X-module, G is a coherent O_{X’}-module, and a : F’ —> G is a map of the completion F’ of F along Z to G whose kernel and cokernel are annihilated by a power of the ideal sheaf of Z. Then there exists a unique coherent O_X-module H with H’ = G and a map F –> H which produces a on completion along Z. This follows from the results on formal glueing which we discussed in this blog post, although this particular statement is a bit easier to prove.

Suppose now that X —> Spec(R) is a proper morphism of schemes with R a Noetherian ring complete wrt an ideal I. Let G be a coherent module on the completion of X along the ideal IO_X. We want to show that G is the completion of a coherent module on X. By Noetherian induction we may assume this is the case whenever G is supported on a proper closed subscheme of X. Chow’s lemma gives U ⊂ X an dense open subscheme and f : Y —> X a U-admissible blowup such that Y is projective over R. By the projective case (which is “easy”) we know that f^*G is the completion of a coherent module H on Y. Let J ⊂ O_X be a quasi-coherent ideal sheaf with Z = V(J) = X – U. Let X’ be the completion of X along Z. By our induction hypotheses the modules G/J^mG are I-adic completions of coherent O_X/J^m modules E_m. The system E_m gives rise to a coherent module E on X’. Then for some n > 0 we obtain a triple (J^nf_*H, E, a) for some map a (this is actually part of Grothendieck’s proof). Applying the result of the previous paragraph we obtain a coherent module F on X. I think it is pretty clear that the completion of F gives G as desired.

What I like about this argument is that it avoids dealing with extensions of formal modules. Note however, that one of the steps of the proof of formal glueing is an Ext computation, so we are not actually avoiding this issue altogether.

Update 10/12/12: Yesterday I finished adding this material to the stacks project. The proof in the projective case is short and sweet, see Section Tag 087V. The proof for the general case is in Section Tag 0886. The exposition avoids working with formal schemes (because it would take several hundred pages to introduce them) and instead consistently works with certain systems of coherent modules. This also has the advantage that the exact same arguments will work in the setting of algebraic spaces (and possibly algebraic stacks).

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?

Grothendieck existence

So I am gearing up to write a bit about Grothendieck’s existence theorem.

Let R be a Noetherian ring complete with respect to an ideal I. Let X be a proper scheme over R. Let O_n = O_X/I^nO_X. Consider an inverse system (F_n) of sheaves on X, such that F_n is a coherent O_n-module and such that the maps F_{n + 1} —> F_n induce isomorphisms F_n = F_{n + 1} ⊗_{O_{n + 1}} O_n. The statement of the theorem is that given any such system there exists a coherent O_X-module F such that F_n ≅ F/I^nF (compatible with transition maps and module structure).

Mike Artin told me Grothendieck was proud of this result.

Because it is all the rage, let’s try to construct F directly from the system via category theory. So consider the functor

G |—-> lim_n Hom_{O_X}(G, F_n)

on QCoh(O_X). Since QCoh(O_X) is a Grothendieck abelian category (see Akhil Mathew’s post) and since this functor transforms colimits into limits, we can apply the folklore result Lemma Tag 07D7. Thus there exists a quasi-coherent sheaf F such that

Hom_{O_X}(G, F) = lim_n Hom_{O_X}(G, F_n)

The existence of F comes for free. (A formula for F is F = Q(lim F_n) where Q is the coherator as in Lemma Tag 077P).

Of course, now the real problem is to show that F is coherent and that F/I^nF = F_n, and I don’t see how proving this is any easier than attacking the original problem. Do you?