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. (more…)