Seminar on Set Theory and Algebra

A friend of mine and I will be running (so to speak) a seminar on set theory and algebra this fall. We will be meeting twice a week (Tuesdays and Thursdays), and the overarching topic will be about the use of set-theoretic methods in solving algebraic problems. I personally will be lecturing once a week, and I wanted to focus on the combinatorial side of things. So I figured that I could use this blog post to address how my half of the lecturing will go.

First of all, my intent is that folks who are familiar with the basics of set theory can actually get something out of this. This means that I will have to introduce quite a bit of set theory as I go along, and perhaps take a number of things for granted. As I go along I’ll try to provide good sources for those sorts of things. With that said, here’s what I want to cover:

Whitehead’s Conjecture: Given an abelian group $A$, we say that $A$ is a Whitehead group (abbreviated W-group) if $Ext^1_\mathbb{Z}(A,\mathbb Z)=0$. Some rather elementary homological algebra gives us that all free groups are W-groups, and Whitehead’s conjecture is that statement that the converse holds. It turns out that whitehead’s conjecture holds for countable groups, so our next concern is groups of size $\aleph_1$. A theorem of Shelah’s tells us that if $V=L$, then Whitehead’s conjecture just holds, while $\mathrm{MA}+\neg\mathrm{CH}$ allows us to construct a non-free W-group of size $\aleph_1$. That is, ZFC cannot decide Whitehead’s conjecture.

I would like to go through the proof of both results. This will require us to discuss $L$, develop some combinatorics in $L$, prove Shelah’s singular compactness theorem, and talk about Martin’s axiom (we will avoid forcing, and take the consistency of $\mathrm{ZFC+MA+\neg CH}$ for granted). Additionally, it gives us a chance to take a look at what generalizations of this problem might look like.

The Vanishing of Ext in L: One direction in which to generalize the above is to look at rings $R$ with left global dimension at most 1. Then given two left $R$-modules $N$ and $M$, we may ask whether or not we can give interesting necessary conditions for $Ext^1_R(N,M)=0$. This turns out to be possible in $L$ (this is due to Eklof), but it requires us to develop a little more combinatorics. In particular we will need to use a consequence of $\square_\kappa$, and a characterization of weakly compact cardinals in $L$ in order to push this result through. One nice thing is that we continue to see many of the ideas from the W-group material pop up again.

Almost Free Abelian Groups: Another generalization is to look at the inductive nature of Shelah’s proof that $V=L$ implies all W-groups are free, and ask whether or not we can push these arguments through under weaker assumptions. That is, suppose $G$ is an abelian group with $|G|>\omega$, and such that every subgroup $H$ of $G$ such that $|H|<|G|$ is free (i.e. $G$ is almost free). When can we show that $G$ itself must be free?

Here we can think of “almost free implies free” as a compactness (in the model-theoretic sense) statement. With that said, it turns out that we have a negative stepping up lemma in the presence of non-reflecting stationary sets. On the other hand, we can’t continue to push the induction through at singular stages, as the singular compactness theorem is a ZFC result. This leads us to ask about successors of singular cardinals, where the question (as usual) turns out to be extremely complicated.

Our exploration of this question will start with $L$, where we can show that we will have almost free, non-free abelian groups at most cardinalities (our issues will arise at weakly compact and singular cardinals). On the other hand, we can still say a lot without the use of $L$. For example, we can show that there is an almost free, non-free abelian group of size $\aleph_{\omega+1}$. Our main tool here will actually be a purely combinatorial statement that is equivalent to the existence of an almost free, non-free abelian group of a particular cardinality. There’s actually quite a bit to say here, but I suspect that I’ll be out of time before I even have a chance to mention some of the more interesting stuff.

I think that I’ll have far too much material to work with here, but starting with Whitehead’s problem allows us a nice introduction to a lot of the main ideas that we would want to explore further. So, even if I don’t get past Whitehead’s conjecture, I won’t feel so bad.