So Bhargav and I were, just earlier today, thinking about the alternating Cech complex in the setting of etale cohomology and this is what we came up with. Caveat: This may be wrong in which case it is my fault (I’m a little worried because the final result of this blog post seems to contradict a throwaway comment in some preprint). Also: it may be in the literature; if you know a reference for this construction please email, thanks.

Let X be an algebraic space. Let U be a separated scheme and let f : U —> X be a surjective etale morphism. Assume that there exists an integer d such that every geometric fibre of f has at most d points. (This is true if U is quasi-compact and X is quasi-separated.) Consider the trace map

f_!Z —> Z

and consider the Koszul complex on this

… —> ∧^2 f_!Z —> f_!Z —> Z

Looking at stalks we see that this is exact. Thus we obtain a quasi-isomorphism K^* —> Z[0] where the complex K^* has as terms K^i = ∧^{i + 1} f_!Z. Moreover, K^i = 0 for i ≥ d. Thus for any abelian sheaf F on X_{etale} we obtain a spectral sequence with E_1-page

E_1^{p, q} = Ext^q(K^p, F)

converging to H^{p + q}(X_{etale}, F). The complex E_1^{*, 0} is our alternating Cech complex.

Now, we want to make explicit the groups Ext^q(K^p, F). These are the right derived functors of Hom(K^p, F). To describe Hom(K^p, F) we introduce some notation. Namely, let W_p be the complement of ALL diagonals in U^{p + 1} = U ×_X … ×_X U. Since f is separated and etale W_p is both open and closed in U^{p + 1}. Moreover, the group S_{p + 1} has a free action on W_p. We claim that

Hom(K^p, F) = S_{p + 1}-anti-invariants in F(W_p)

To see this look at (W_p —> X)_!Z. The stalk of this sheaf at a geometric point x of X is the free Z-module with basis the set of injective maps {0, …, p} —> U_x. Hence ∧^{p + 1}f_!Z is the maximal S_{p + 1}-anti-invariant quotient of (W_p —> X)_!Z. This proves the displayed formula. Since S_{p + 1} acts freely on W_p over X the quotient U_p = W_p/S_{p + 1} is an algebraic space etale over X. There is a way to “twist” F|_{U_p} by the sign character S_{p + 1} —> {+1, -1} giving a sheaf F_p on U_p. With a little bit of work we obtain

Ext^q(K^p, F) = H^q(U_p, F_p).

Why is this useful? Suppose that F is a quasi-coherent O_X-module, X is quasi-compact, X is separated, and U is affine. Then each W_p is affine too, and so is U_p. Moreover, the sheaves F_p are still quasi-coherent. Thus we see that the E_1^{p, q} are nonzero only when q = 0 and we obtain vanishing of H^n(X, F) for all n >= d! This is exactly the vanishing you traditionally obtain from the alternating Cech complex associated to a finite affine open covering of a scheme.

For a quasi-compact, quasi-separated algebraic space X we can redo the argument with U an affine scheme. We find (because we can apply the previous result to the separated quasi-compact algebraic spaces U_p) that X has finite cohomological dimension for quasi-coherent sheaves. And that’s the thing I was stuck on in the stacks project yesterday…

[Edit Aug 19, 2011: This material is now in the stacks project. The spectral sequence is Lemma Tag 0728. The application to algebraic spaces is Proposition Tag 072B and Lemma Tag 072C. Note that the first vanishing result is interesting for schemes also.]

Some types of questions in algebra immediately reduce to the “countable” case. Simple example: Let A be a ring and let φ : A —> A be an automorphism. Then for every finite subset E of A there exists a countable subring A’ ⊂ A containing E such that φ induces an automorphism of A’. The proof is to let A’ be the subring of A generated by φ^n(e) for all e in E and all n ∈ Z.

Another example of this phenomenon is that any projective module is a direct sum of countably generated (projective) modules (Kaplansky’s theorem).

The technique also applies to the following problem: Let A ⊂ B be an integral extension of rings with A Noetherian. Let M be an A-module such that M ⊗_A B is flat over B. Problem: Show M is flat over A. (This is equivalent to the direct summand conjecture by a 1 page paper of Ohi.) A key case is to show M ⊗_A B = 0 implies M = 0. Picking suitable families of elements this reduces to the case where B is countably generated over A and M is a countably generated A-module.

Here are two examples involving algebraic stacks: (1) Suppose X is a quasi-compact algebraic stack with affine diagonal. I claim you can write X as a filtered limit of stacks X’ of the form X’ = [U’/R’] with U’ and R’ spectra of countable rings. I haven’t written out the details but it seems to me one can do this by just “adding elements” as above. (2) A quasi-coherent module F on a quasi-separated and quasi-compact algebraic stack is a filtered colimit of countably generated quasi-coherent modules.

I wonder if this type of argument can ever be used to bootstrap? Do some general arguments become easier if you assume all rings/modules in question are countable? Are there some _useful_ properties that hold for countable rings? Things like “the topology on Spec has a countable basis” aren’t really useful, or are they?

Before you say “No!” let me just point out that Kaplansky’s theorem is used in the proof of faithfully flat descent for projectivity of modules, so sometimes…

Let X = Spec(A) be an affine scheme. Let M, N be A-modules. Let F, G be the sheaves of O_{big}-modules on the big fppf site of X associated to M and N, e.g., F(Spec(B)) = B ⊗_A M and similarly for G. As a by-product of the material on adequate modules I proved the following formula

Ext^i_{O_{big}}(F, G) = Pext^i_A(M, N)

The ext group on the left is the ext group in the category of all O_{big}-modules. The ext group on the right is the ith pure extension group of M by N over A. This group is computed by taking a universally exact resolution 0 -> N -> I^0 -> I^1 -> … with each I^j pure injective and taking the ith cohomology group of Hom_A(M, I^*). An A-module I is pure injective if for any universally injective map M_0 -> M_1 the map Hom_A(M_1, I) -> Hom_A(M_0, I) is surjective.

There seems to be a lot of papers on pure modules, pure injectivity, etc. Gruson and Jensen characterized pure injective modules as those modules such that the functor – ⊗_A M : mod-A —> Ab is injective in the functor category (mod-A, Ab). Here mod-A is the category of finitely presented A-modules. It follows that

Ext^i_{(mod-A, Ab)}(- ⊗_A M, – ⊗_A N) = Pext^i_A(M, N)

Our formula above is about O_{big}-modules, which in terms of functors means functors F : Alg_A —> Ab such that F(B) has the structure of a B-module for every A-algebra B and such that B —> B’ gives a B-linear map F(B) —> F(B’). These are called module-valued functors (terminology due to Jaffe). Then we can rewrite the first equality above as

Ext^i_{module-valued functors}(F, G) = Pext^i_A(M, N)

where F(B) = B ⊗_A M and G(B) = B ⊗_A N. In this formula you can let Alg_A be any sufficiently large category of A-algebras, e.g., the category of finitely presented A-algebras.

The two results seem related. But there is a big difference between the functor categories (mod-A, Ab) and (Alg_A, Ab). Namely, if we look at Ext^i_{(Alg_A, Ab)}(F, G) then we get a completely different animal. For example suppose that G_a(B) = B for all A-algebras B and suppose that A is an F_p algebra. Then we see that Hom_{(Alg_A, Ab)}(G_a, G_a) contains the frobenius map frob : G_a —> G_a which on values over B raises every element to the pth power. In fact, the work of Breen on ext groups of abelian sheaves on the fppf-site (warning: this is not exactly what he studies there) implies some of the higher ext groups Ext^i_{(Alg_A, Ab)}(G_a, G_a) are nonzero also (lowest case seems to be i = 2p)!

Conclusion: module-valued functors over Alg_A and abelian group valued functors on mod-A somehow ends up giving the same ext groups for the functors associated to A-modules described above.

During the last few weeks I have been working on a way to described the category of quasi-coherent modules on a scheme X in terms of the big fppf site of X. I think I have succeeded to some extend, and I’d like to explain some of the results here. But first, let me say why this may be (somewhat) useful.

Let us denote O_X the structure sheaf on the scheme X and O_{big} the structure sheaf on the big fppf site of X. Similarly, given a morphism f : X —> Y of schemes we have the usual pushforward f_* and the pushforward f_{big, *} on sheaves on the big sites. My goal was to understand the following two phenomena:

1. The fully faithful embedding i_X : QCoh(X) —> Mod(O_{fppf}) isn’t exact in general.
2. When f : X —> Y is quasi-compact and quasi-separated, then f_{big, *}i_X(F) is not equal to i_Yf_*F in general.

Of course for schemes this isn’t a problem, but 1 and 2 also happen for algebraic stacks where we do not have the luxury of an underlying ringed space whose category of quasi-coherent modules is the “right one”. This means that in order to define pushforward for quasi-coherent modules (along quasi-compact and quasi-separated maps) one has to be a little bit careful (it isn’t hard — I’ll come back to this). Secondly, as was pointed out to me several times by Martin Olsson, the first problem means that D_{i_X(QCoh)}(O_{big}) isn’t a triangulated subcategory of D(O_{big}). This isn’t a problem for schemes because you can take D_{QCoh}(O_X) but again this doesn’t work for algebraic stacks and you have to do something. A solution for this second problem is to work with the lisse-etale site, but then you get embroiled in the nonfunctoriality of it…

OK, so I have a “solution” to these two problems. Let me say right away that the solution isn’t ideal, partly because it is rather complicated. But at the end of the story (after a certain amount of work) the picture that emerges is rather pleasing.

First assume that X is an affine scheme. It turns out that problem 1 for affine X was solved in a paper by Jaffe entitled Coherent Functors, with Application to Torsion in the Picard Group. In this wonderful paper he points out that if you just add kernels of maps of quasi-coherent O_{big}-modules, then you obtain an abelian subcategory of D(O_{big})! I’m going to call these adequate modules (these correspond to the “module-quasi-coherent A-functors” of Jaffe’s paper). On a general scheme X I am going to say an O_{big}-module F is adequate if there exists an affine Zariski open covering of X such that F restricts to adequate modules over the members of the covering. It moreover turns out that adequate modules are preserved under colimits and that they form a Serre subcategory of the abelian category all O_{big}-modules.

You can show quite easily that one has vanishing of cohomology of adequate modules over affines, so that they behave much like quasi-coherent modules. Moreover, any quasi-coherent module is adequate and (Zariski) locally any adequate module is a kernel of a map of quasi-coherent modules. Finally, if you have an adequate module on X and you restrict it to the small Zariski-site of X then you get a quasi-coherent module. This implies readily that

QCoh(X) = Adeq(X)/C(X)

where C(X) is the category of parasitic adequate modules. An adequate O_{big}-module is called parasitic if the restriction to S_{Zar} is zero (it was a suggestion by Martin Olsson that these should play an important role in the story).

So this solves problem 1 as Adeq(X) —> Mod(O_{big}) is exact by construction and in fact Adeq(X) is a Serre subcategory. What about 2? The answer is that R^if_{big, *}F is adequate for adequate modules F when f : X —> Y is quasi-compact and quasi-separated. Moreover, R^if_{big, *}F is in C(Y) if F is in C(X). Hence we see that R^if_{big, *} induces a functor

QCoh(X) = Adeq(X)/C(X) —-> Adeq(Y)/C(Y) = QCoh(Y)

and (you guessed it) this recovers our usual R^if_* for quasi-coherent sheaves! Thus a solution to 2.

Morally, this tells us that we should view QCoh(X) as a subquotient of Mod(O_{big}) and not as a subcategory. Taking a quotient of an abelian category by a Serre subcategory is achieved by Gabriel-Zisman localization. This suggests that we can do the same with derived categories. Indeed, it turns out that

D_{QCoh(X)}(O_X) = D_{Adeq(X)}(O_{big})/D_{C(X)}(O_{big})

(Verdier quotient) with no conditions on X whatsoever. I expect a description of the total direct image Rf_* on the left hand side as the functor induced by Rf_{big, *} on D_{Adeq}(O_{big}) in exactly the same way as above (details not yet written).