The Gabriel-Popescu theorem and a variant of Kuhn

This post is part of a series on the Sullivan conjecture in algebraic topology. The Sullivan conjecture is a topological result, which remarkably reduces — as H. Miller showed —  to a purely algebraic computation in the category of unstable modules (and eventually algebras) over the Steenrod algebra, and in particular an injectivity assertion. This is a rather formidable category, but work of Kuhn enables one to identify a quotient category of it with the category of “generic representations” of the general linear group, which can be studied using different (and often easier) means. Kuhn’s work provides an approach to proving much of the algebraic background that goes into the Sullivan conjecture. In this post, I’ll describe one of the important ingredients.

The Gabriel-Popsecu theorem is a structure theorem for Grothendieck abelian categories, a version of which will be useful in understanding the structure of the category of unstable modules over the Steenrod algebra. The purpose of this post is to discuss this result and its many-object version due to Kuhn, from the paper “Generic Representations of the Finite General Linear Groups and the Steenrod Algebra: I.” Although the proof consists mostly of a series of diagram chases, there are some subtleties that I found rather difficult to grasp, and I thought it would be worthwhile to go through it in detail here.

1. Grothendieck abelian categories

Let be an abelian category. Then is Grothendieck abelian if

• has a generator: that is, there is an object such that every object in can be built up from colimits starting with . (More precisely, the smallest subcategory of , closed under colimits, that contains is itself)
• Filtered colimits in exist and are exact.

Many categories occurring in “nature” (e.g., categories of modules over a ring of sheaves on a site) are Grothendieck, and it is thus useful to have general results about them. The goal of this post is to describe a useful structure theorem for Grothendieck abelian categories, which will show that they are the quotients of categories of modules by Serre subcategories.

 

Quasi-coherent sheaves, presentable categories, and a result of Gabber

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.