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.<\/p>\n
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:<\/p>\n
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…<\/p>\n
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.<\/p>\n
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<\/a>. 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<\/em> (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.<\/p>\n 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<\/p>\n QCoh(X) = Adeq(X)\/C(X)<\/p><\/blockquote>\n where C(X) is the category of parasitic adequate modules. An adequate O_{big}-module is called parasitic<\/em> 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).<\/p>\n 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<\/p>\n QCoh(X) = Adeq(X)\/C(X) —-> Adeq(Y)\/C(Y) = QCoh(Y)<\/p><\/blockquote>\n and (you guessed it) this recovers our usual R^if_* for quasi-coherent sheaves! Thus a solution to 2.<\/p>\n 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<\/p>\n D_{QCoh(X)}(O_X) = D_{Adeq(X)}(O_{big})\/D_{C(X)}(O_{big})<\/p><\/blockquote>\n (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).<\/p>\n","protected":false},"excerpt":{"rendered":" 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, … Continue reading