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 , one can consider the category of quasi-coherent sheaves on . This is an abelian category, a subcategory of the category of all sheaves on . 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 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 is presentable. Then, since filtered colimits are exact in , it will follow that is cocomplete. Here is the result of Gabber:

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

Here having “rank ” means that a quasi-coherent sheaf is locally generated by sections. For a quasi-compact, quasi-separated scheme, we can take , 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 is quasi-coherent, then commutes with **huge** colimits: this is a consequence of the smallness of every module over a ring. Moreover, given , there can only be so many—a small set—of quasi-coherent sheaves of rank on . It follows that 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 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

from quasi-coherent sheaves on to sheaves of modules over the structure sheaf. This is a colimit-preserving functor, and since is *presentable,* as we have just seen, the adjoint functor theorem gives a right adjoint

In other words, given any -module on , there is a quasi-coherent sheaf together with a map which is *universal* from maps into from quasi-coherent sheaves. [*Fixed an error here.*]

But now it’s clear how to take limits in : just take the limit in , and apply !

July 31, 2011 at 4:02 am

While your argument which explicitely constructs all limits, is nice, it is not really needed: By the adjoint functor theorem or some version thereof any Grothendieck category is complete.

July 31, 2011 at 6:33 pm

Interesting. How does the argument work?

(By the way, I can’t take credit for any of the arguments in the post: I learned them from mostly Theo Buehler’s post, and apparently he learned them from an article by Thomason and Trobaugh.)

December 4, 2011 at 5:06 pm

There’s an argument for completeness of Grothendieck categories (henceforth G in this message) via the Gabriel-Popescu theorem: it says that your category embeds fully faithfully into a module category by a functor whose left adjoint is exact. Full reflective subcategories of complete categories are again complete, so that’s that.

December 4, 2011 at 6:52 pm

Thanks for pointing this out.

July 31, 2011 at 1:46 pm

Akhil why not talk about more applied topics from time to time?

August 3, 2011 at 9:24 am

Akhil — Regarding the previous comment, I really enjoy the topics you talk about. Please don’t change this to an applied math blog.

August 3, 2011 at 12:52 pm

Thanks for the encouragement. I certainly didn’t have any plans to change the direction of this blog.

October 16, 2011 at 10:54 am

[…] as crazy as it sounds, e.g., the category of quasi-coherent sheaves on a scheme has products, see Akhil Mathew’s post. (In any case being parasitic is preserved under products.) Hmm? I’ll think more. This […]

October 16, 2011 at 7:35 pm

[…] is a follow-up to Akhil Mathew’s blog post which explains that the category of quasi-coherent sheaves on a scheme X is a Grothendieck abelian […]

September 7, 2012 at 10:06 am

[…] QCoh(O_X). Since QCoh(O_X) is a Grothendieck abelian category (see Akhil Mathew’s post) and since this functor transforms colimits into limits, we can apply the folklore result Lemma Tag […]