This post is a followup on this series of posts. Basically, I in some sense forgot the punchline of the whole story and just now it came back to me.

Let’s consider the case of the Quot functor for example. First let’s consider it in a reasonable level of generality: assume we have a proper morphism X —> S with S Noetherian and a coherent O_X-module F flat over S. Then given a Noetherian S-algebra A and a short exact sequence

0 —> E —> F_A —> G —> 0

on X_A = X ×_S Spec(A) with G flat over A the obstructions to deforming this to a thickening A’ of A lie in Ext^1_{O_X}(E, G ⊗ I) and if the obstruction vanishes the set of deformations is principal homogeneous under Ext^0_{O_X}(E, G ⊗ I).

In order to apply Artin’s method to get openness of versality exactly as in Section 07YV we need to prove there exists a perfect complex of A-modules K such that we have

Ext^1_{O_X}(E, G ⊗ M) = H^2(K ⊗_A M)

and

Ext^0_{O_X}(E, G ⊗ M) = H^1(K ⊗_A M)

functorially in the A-module M and agreeing with boundary maps. What is important here is that you only need to have this for low cohomology groups.

**Guess:** There exists a bounded below complex of finite projective A-modules P and functorial isomorphisms Ext^i_{O_X}(E, G ⊗ M) = H^{i + 1}(P ⊗_A M).

Firstly, if the guess is true, then we get what we want by just taking K to be a *stupid* truncation of P!

Secondly, if X —> S is projective, then the guess is correct. Namely, in this case you can find a resolution

…—> E_1 —> E_0 —> E —> 0

of E where each E_i is a finite direct sum O_X(-n) where n ≫ 0. Then the complex

P = (f_*Hom(E_0, G) —> f_*Hom(E_1, G) —> … )

works because each of the flat sheaves Hom(E_i, G) will have only cohomology in degree 0 and hence f_*Hom(E_i, G) is finite locally free on Spec(A). Moreover, the complex will compute the correct cohomology after tensoring with M. (Observation: you do not need E to be flat over A for this argument to work; the key is that G is flat over A.)

Thirdly, in general, if we can find a perfect complex E’ on X and a map E’ –> E which is a quasi-isomorphism in degrees > -5 (for example), then in order to compute Ext^i(E, G ⊗ M) for i < 4 (or something) we can replace E by E’. This leads us to something like Rf_*Hom(E’, G) which is a perfect complex on Spec(A) by standard arguments. Grothendieck uses this argument to study Ext^0 in EGA III, Cor 7.7.8. Anyhow, this should allow us to handle the case where you have an ample family of invertible sheaves on X (that’s not much better than the projective case of course). [**Edit:** Jack just emailed that the existence of E’ (in the case of schemes) is in a paper by Lipman and Neeman entitled “QUASI-PERFECT SCHEME-MAPS AND BOUNDEDNESS OF THE TWISTED INVERSE IMAGE FUNCTOR”.]

Fourthly, it may well be that the guess follows from Jack Hall’s paper “Coherence results for algebraic stacks”. It is obviously very close and it may just be a translation, but I haven’t tried to think about it. I’d like to know if this is so. [**Edit:** Jack just explained to me that this is true for example if the base is of finite type over Z which we can reduce to I think.]

Fifthly, if F is not assumed flat over S, then we should probably do something like look at distinguished triangles

E —> F ⊗^**L** A —> G —> E[1]

where now E is an object of D(X_A) but G is a usual O_{X_A}-module (placed in degree 0) flat over A and hopefully the obstructions lie in Ext^1(E, G ⊗ I), etc, and we can try imitating the above. This is just pure speculation; I’ll have to check with my friends to see what would be the correct thing to do. Just leave a comment if you know what to do.

I think your “fifthly” is roughly what I suggested in my earlier comment: replace O_X by a simplicial resolution of f^{-1}O_S-algebras that are termwise flat (or even free), and similarly replace F by a simplicial module over this simplicial f^{-1}O_S-algebra that is termwise flat.

OK, yes, I thought so. I didn’t want to mention your name, since I wanted you to be able to deny having said anything like this. Doing the simplicial thing might be a good idea. Thanks.

Oops, I forgot it was an e-mail, not a post.

Just curious, will the stacks project include Chow group for Deligne-Mumford stacks and things like the virtual fundamental class?

OK, so we have the chapter Chow homology and Chern classes. This was written a while back. It defines Chow groups of schemes locally of finite type over a fixed locally Noetherian universally catenary base S endowed with a dimension function without imposing separation conditions. The idea was that working in this level of generality would give some insight into the problems that come up when defining Chow groups for algebraic stacks. This chapter needs some more work (see the todo list). Once this is done a lot of what is written there should also work in the setting of Deligne-Mumford stacks, but obviously this would require a lot of work.

On the other hand, it is perhaps more interesting to think about Chow groups of Artin stacks (locally of finite type over S as before). Here we have Kresch’s approach which isn’t ideal but is I think the best thing we have right now in the sense that it works (possibly with additional work) for the widest possible class of stacks. So in some sense this area still has ongoing research.

The virtual fundamental class thing is something I certainly would want to see discussed sometime. There already is a huge literature on this (even just working out the basics of the thing). It is going to be a nontrivial task to start discussing it. It is not even clear to me what would be the right level of generality to discuss this in for example.

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