I’ve been reading an interesting paper of Adams, Haeberly, Jackowski, and May on the Atiyah-Segal completion theorem. One of the surprising features of this paper is the heavy use of pro-abelian groups to deal with the inconvenient fact that inverse limits are generally not exact in abelian groups. I’d like to blog about the proof in this paper, but first I’d like to go through some of the background on pro-objects. In this post, I’ll describe the entirely dual picture of ${\mathrm{Ind}}$-objects, which is (at least for me) easier to understand.

1. Definition

Let ${\mathcal{A}}$ be a small abelian category. Then there is an imbedding

$\displaystyle \mathcal{A} \hookrightarrow \mathrm{Ind}(\mathcal{A}),$

of ${\mathcal{A}}$ into the larger category of ind-objects of ${\mathcal{A}}$. One benefit of doing this is that ${\mathrm{Ind}(\mathcal{A})}$ is a larger abelian category containing ${\mathcal{A}}$, in which there are enough injectives.

I always found the traditional definition of these confusing, so let me describe another definition (which happens to generalize nicely to the ${\infty}$-categorical case, and which is where I learned it from).

Let ${\mathcal{C}}$ be any category. Then we know that the category ${P(\mathcal{C}) = \mathrm{Fun}(\mathcal{C}^{op}, \mathbf{Sets})}$ is the “free cocompletion” of ${\mathcal{C}}$: that is, given any cocomplete category ${\mathcal{D}}$, we have an equivalence

$\displaystyle \mathrm{Fun}(\mathcal{C}, \mathcal{D}) \simeq \mathrm{Fun}^{L}( P(\mathcal{C}), \mathcal{D})$

between functors ${\mathcal{C} \rightarrow \mathcal{D}}$ and colimit-preserving functors ${P(\mathcal{C}) \rightarrow \mathcal{D}}$. The ${\mathrm{Ind}}$-category is defined to have an analogous universal property, except that one just takes filtered colimits. (more…)

The story of how I ended up writing this post is a bit roundabout. I was trying to figure out whether a technical lemma in Hartshorne on the cohomology of inductive limits of sheaves could be proved using spectral sequences, and I thought I had it, but I then realized that I had no justification for asserting that the category in question had enough injectives (which is necessary to apply the Grothendieck spectral sequence). So I tracked down the original reference to the result, which was–conveniently enough–in Grothendieck’s famous Tohoku paper (available openly).  It turns out that there is a way to see that the category I was interested in had enough injectives, but it is a fairly interesting and involved result. I will explain what I learned from reading the section of the paper today. Next time, I will explain my thoughts that led me here. Also, before I proceed, here’s a PDF of the post.

It is known that the category of modules over a ring has enough injectives, i.e. any object can be imbedded as a subobject of an injective object. This is one of the first things one learns about injective modules, though it is a nontrivial fact and requires some work. Similarly, when introducing sheaf cohomology, one has to show that the category of sheaves on a given topological space has enough injectives, which is a not-too-difficult corollary of the first fact.

However, it is more difficult to see that, for instance, the category of inductive diagrams of sheaves (within a fixed inductive system) has enough injectives. For this a general result that states that wide classes of abelian categories (satisfying minor categorical conditions) have enough injectives is handy and convenient. This result is in Tohoku, but Grothendieck claims it is not his.

I will assume familiarity with basic diagram-chasing in abelian categories (e.g. the notion of the inverse image of a subobject). (more…)

The previous post got somewhat detailed and long, so today’s will be somewhat lighter. I’ll use completions to illustrate a well-known categorical trick using finite presentations.

The finite presentation trick

Our goal here is:

Theorem 1  Let ${A}$ be a Noetherian ring, and ${I}$ an ideal. If we take all completions with respect to the ${I}$-adic topology,

$\displaystyle \hat{M} = \hat{A} \otimes_A M$

for any f.g. ${A}$-module ${M}$.   (more…)