A short post on Generators

So one of my goals for this semester (or year) is to try and figure out what’s going on in Section 2 of Sh460. Of course, the section starts off by referencing Claim 6.7A of Sh430 and improving it (without mentioning what’s actually going on in that claim). Looking back at Claim 6.7A of Sh430, it turns out that this references some of the tools used in the proof of Claim 6.7, which gives us the existence of closed and transitive generators. Now it turns out that one of the things that we worked through in the summer school at UC Irvine (which I like to call pcf-fest 2016) is this very thing.

The proof that James gave was a bit different, but I think that claim 6.7A is really just making more explicit the relationship between transitive generators, universal sequences, and $\kappa$-IA elementary substructures. So what I’d like to do first is go back and work through the existence of transitive generators, and see how much of this stuff I can tease out along the way. Hopefully that’ll also put me in a good mindset to work through the Sh460 stuff. I figured that a good place to start is with the usual construction of generators and how they relate to universal sequences.

Throughout this, I’m going to let $A$ be a collection of regular cardinals, and put restrictions on it as necessary.

Definition: Let $A$ be a set of regular cardinals, and define

$pcf(A)=\{cf(\prod A, <_U): U\text{ is an ultrafilter on }A\}$

Here $<_U$ is just domination modulo $U$. I will frequently bounce between $(\prod A, <_U)$ and $\prod A/U$.

Definition: Let $\lambda$ be a regular cardinal, then

$J_{<\lambda}[A]=\{X\subseteq A : pcf(X)\subseteq\lambda\}$

Note that this is an ideal on $A$.

Definition: We say that $B_\lambda$ is a generator of $J_{<\lambda}$ if $\lambda\in pcf(A)$, and $J_{<\lambda^+}[A]$ is generated from $J_{<\lambda}[A]$ by $B_\lambda$.

In particular, we see that $J_{<\lambda^+}[A]=\{X\subseteq A : X\subseteq_{J_{<\lambda}[A]}B_\lambda\}$. Also note that if $\lambda\notin pcf(A)$, then obviously $pcf(X)\subseteq\lambda^+\implies pcf(X)\subseteq \lambda$ for $X\subseteq A$. So in the case that $\lambda\notin pcf(A)$, asking for a generator is fantastically uninteresting. Now, let’s say that $A$ is progressive whenever $|A|^+<\min(A)$.

Definition: Let $\lambda\in pcf(A)$, then $\vec f^\lambda=\langle f^\lambda_\alpha : \alpha<\lambda\rangle$ is a universal sequence for $\lambda$ if:

1. $\vec f^\lambda$ is $<_{J_{<\lambda}}$-increasing;
2. For any ultrafilter $U$ over $A$ such that $cf(\prod A/U)=\lambda$, we have that $\vec f^\lambda$ is cofinal in $\prod A/U$.

Note that if $U$ is an ultrafilter with $cf(\prod A/U)=\lambda$, then $U\cap J_{<\lambda}=\emptyset$. Otherwise, there is some $X\subseteq A$ with $pcf(X)\subseteq\lambda$ and $X\in U$. But then, $cf(\prod X/U)=cf(\prod A/U)$ since $X=_U A$, which would mean that $\lambda\in pcf(X)$. This gives us another characterization of $J_{<\lambda}[A]$ as the collection of subsets of $X$ which forces $cf(\prod A/U)<\lambda$ whenever they get assigned measure one by $U$.

Theorem (Shelah): If $A$ is progressive, then for every $\lambda\in pcf(A)$, there is a universal sequence for $\lambda$ with a $J_{<\lambda}$ exact upper bound $h\in{}^A ON$.

Why are universal sequences useful? Well, if $f^\lambda$ is a universal sequence for $\lambda \in pcf(A)$ with exact upper bound $h$, then the set $B_\lambda=\{a\in A : h(a)=a\}$ is actually a generator for $\lambda$. Now, these generators are only unique modulo $J_{<\lambda}$, and so we have some room to massage them. In the next post, I want to examine the possibility of doing just that.