Challenge

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Let f : X —> B be a morphism of algebraic spaces. Let u : F —> G be a map of quasi-coherent O_X-modules. Consider the functor

F : (Sch/B) —> (Sets), T |—> singleton if u_T is zero and empty else

This functor always satisfies the sheaf property for the fpqc topology (Lemma 083H). It turns out that if f is locally of finite presentation, G is locally of finite presentation, G is flat over B, and the support of G is proper over B, then F is an algebraic space and F —> B is a closed immersion. This is Lemma 083M and the proof uses the Raynaud-Gruson techniques.

Challenge: Give a simple proof of Lemma 083M.

A while back I tried to do this. First, some reductions: you can reduce to the case where B is an affine scheme. You can reduce to the case where f is proper and locally of finite presentation (replace F by image of F in G and replace X by suitable closed subspace supporting G). I think you can also reduce to the case where F is of finite presentation (by a limit argument). Hence, if you like, you can reduce to the case where B is the spectrum of a Noetherian ring and everything is of finite type.

In the case B is affine there is a simple argument that shows: if u_T = 0 for some quasi-compact T over B, then there is a closed subscheme Z ⊂ B such that u_Z = 0 and such that T —> B factors through Z (Lemma 083K). The proof only uses that G is flat over B.

The problem left over is somehow: What if we have infinitely many closed subschemes Z_1, Z_2, Z_3,… ⊂ B such that u_{Z_i} is zero. Why is it true that u_Z = 0 where Z is the scheme theoretic closure of ⋃ Z_n? E.g., what if B = Spec(Q[x, y]) and Z_n is cut out by the ideal (x^n, y – 1 – x – x^2/2 – x^3/6 – … – x^n/n!).

If F is globally generated then you can reduce to the case F = O_X and you can use that Rf_*G is (universally) computed by a perfect complex. This is related to Jack Hall’s paper “Coherence results for algebraic stacks”. Note that Lemma 083M is a consequence of the results there. Jack’s paper uses relative duality which we do not have available in the Stacks project.

If f is projective, you can reduce to the case F = O_X and G such that Rf_*G is universally computed by a finite locally free sheaf, whence the result. This case is straightforward using only standard results.

If the support of G is finite over B then the result is elementary. So you could try to argue by induction on the relative dimension. Alas, I’m having trouble producing enough quotients of G which are flat over B.

I still think something simple might work in general. But I don’t see it. Do you?

11 thoughts on “Challenge”

  1. Dear Johan, Theorem 2.1 of my preprint with Mike Roth,
    “Weak Approximation and R-Equivalence over Function Fields of Curves”, available at http://www.math.sunysb.edu/~jstarr/papers/wa0809a.pdf. In the projective case, this is in EGA III, Cor. 7.7.8. Also, generally, it follows pretty quickly from Prop. 2.1.3 of Lieblich’s “Remarks on Coherent Algebras …”.
    In fact, in an early draft of my joint paper with you on rationally simply connected varieties, we were planning to use Theorem 2.1. Perhaps you have weakened one of the hypotheses and I missed it, but I believe your challenge follows immediately from that result. Best, Jason

    • OK, thank you Jason! I should have looked there for sure… and now I have:

      The proof in Lieblich (of Prop 2.3, there is no 2.1.3 in the published version) is “not simple” in the sense that it uses a lot of theory (in some sense it proceeds by verifying Artin’s axioms for the functor). This is not “simple” enough: I already have a proof which doesn’t use Popescu’s theorem or Artin’s approximation.

      In EGA III, 7.7.8 it is assumed that G is a cokernel of a map of finite locally free modules. It is more general then being globally generated, but not much. This is presumably why you write that it applies to give the projective case (I agree).

      • Johan, Out of curiosity, can you prove the representability of Hom(F,G) using the same methods? I was also “vexed” by the fact that one needs Artin’s theorems to prove the representability of Hom(F,G) in this context. If there is an elementary proof of that, I would like to see it. Best, Jason

        • Dear Jason, Perhaps you are aware of this, but an immediate corollary of Jack Hall’s coherence result (arXiv:1206.4179, Thm A) is that Hom(F,G) is representable by an affine scheme – indeed, even by Spec(Sym(module)). This is Thm B (loc. cit.). I noticed that you also prove this in your preprint with Mike Roth after first proving that it is representable by an algebraic space (using either the stack Coh or the space Quot).

          Jack is not using representability of Coh/Quot nor Artin algebraization or Néron-Popescu desingularization. The current proof (loc. cit.) is using dualizing complexes though. There are at least two other proofs, also due to Jack, that avoids dualizing complexes.

          1) The first uses perfect generation of the unbounded derived category and applies to algebraic stacks with quasi-finite diagonals (arguably, this fact is at least as complicated as dualizing complexes).

          2) The second alternative proof is a somewhat complicated but yet rather standard dévissage that I don’t think involves more than Chow’s lemma and some basic homological/commutative algebra, thus “elementary”.

          I think both these proofs will be included in a revised version of 1206.4179. As a bonus, using coherent functors, one also obtains representability of Hom(F,G) for arbitrary quasi-coherent F (not necessarily of finite presentation).

  2. Just to say one work about how to apply Theorem 2.1, or the related theorems in the other sources: the morphism u defines a morphism from the representing module N to the structure sheaf of X. The image of this morphism defines the ideal sheaf of the closed subscheme representing your functor F.

    • OK, but the proof uses the fact that Quot is a scheme (in my setting just an algebraic space) which I wouldn’t call “elementary”. My hope is that Lemma 083M could really have an elementary proof.

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