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).