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.

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