So I’ve failed in my duty as a math blogger. I actually have been writing stuff up, notes on my project, but most of them are not suitable (e.g., too detailed) for a blog. What I really should be doing is blogging about more elementary stuff. I have been trying to learn about Kähler manifolds lately; maybe I can start a short series on them.

Today, I would like to point an interesting observation due to Gabber, which I learned about through a discussion with Theo Buehler on math.SE. Given a scheme {X}, one can consider the category {\mathrm{Qco}(X)} of quasi-coherent sheaves on {X}. This is an abelian category, a subcategory of the category of all sheaves on {X}. Moreover, it is closed under colimits. Now the latter category is wonderfully nice: it’s a Grothendieck abelian category. In other words, it has a system of generators and filtered colimits are exact. One consequence of being a Grothendieck abelian category is that there are automatically enough injectives, by a result in the famous “Tohoku” paper.

But the observation of Gabber shows that {\mathrm{Qco}(X)} is a complete, cocomplete Grothendieck abelian category too. There are a few reasons one might care about this. For one, Grothendieck abelian categories are presentable categories. Basically, presentability means that arguments such as Quillen’s small object argument in homotopy theory work out: namely, one has a “small” set of “compact” objects that generates the category under colimits. (This is in fact the idea behind Grothendieck’s proof that such a category has enough injectives.) Moreover, and perhaps more importantly in this case, the adjoint functor theorem becomes nicer for presentable categories.

Recall that a left adjoint functor between categories preserves colimits, and a right adjoint preserves limits. The adjoint functor theorems (and there are various incarnations thereof) state that, under suitable set-theoretic conditions, the converse holds as well. One example is the Brown representability theorem for triangulated categories, which states that commuting with coproducts is a sufficient condition for a triangulated functor between nice triangulated categories to be a left adjoint. Under presentability hypotheses, the adjoint functor theorem also goes into effect (this could make another blog post!).

Well, so how are we going to show all this? Actually, the strategy is to show first that {\mathrm{Qco}(X)} is presentable. Then, since filtered colimits are exact in {\mathrm{Qco}(X)}, it will follow that {\mathrm{Qco}(X)} is cocomplete. Here is the result of Gabber:

Theorem 1 Given a scheme {X}, there is a cardinal {\kappa} such that any quasi-coherent sheaf on {X} is the filtered colimit of its quasi-coherent subsheaves of rank {\leq \kappa}.

Here having “rank {\leq \kappa}” means that a quasi-coherent sheaf is locally generated by {\leq \kappa} sections. For a quasi-compact, quasi-separated scheme, we can take {\kappa = \aleph_0}, by a result in EGA I.9. This is also an exercise in Hartshorne in the noetherian case: a quasi-coherent sheaf on a noetherian scheme is the filtered colimit of its coherent subsheaves.

So how does this give us presentability? Well, any quasi-coherent sheaf is small in that if {\mathcal{F}} is quasi-coherent, then {\hom(\mathcal{F}, \cdot)} commutes with huge colimits: this is a consequence of the smallness of every module over a ring. Moreover, given {\kappa}, there can only be so many—a small set—of quasi-coherent sheaves of rank {\leq \kappa} on {X}. It follows that {\mathrm{Qco}(X)} is generated by a small set of compact objects, so is a presentable abelian category.

Now, by Grothendieck’s theorem, it is already clear that {\mathrm{Qco}(X)} has enough injectives (i.e. is a Grothendieck abelian category). But we can even show more: it is a complete category. To see this, consider the inclusion functor

\displaystyle \mathrm{Qco}(X) \rightarrow \mathrm{Sh}_{\mathcal{O}_X}(X),

from quasi-coherent sheaves on {X} to sheaves of modules over the structure sheaf. This is a colimit-preserving functor, and since {\mathrm{Qco}(X)} is presentable, as we have just seen, the adjoint functor theorem gives a right adjoint

\displaystyle R: \mathrm{Sh}_{\mathcal{O}_X}(X) \rightarrow \mathrm{Qco}(X).

In other words, given any {\mathcal{O}_X}-module {\mathcal{M}} on {X}, there is a quasi-coherent sheaf {R(\mathcal{M})} together with a map {R(\mathcal{M}) \to \mathcal{M} } which is universal from maps into {\mathcal{M}} from quasi-coherent sheaves. [Fixed an error here.]

But now it’s clear how to take limits in {\mathrm{Qco}(X)}: just take the limit in {\mathrm{Sh}_{\mathcal{O}_X}(X)}, and apply {R}!