Downward categoricity from a successor inside a good frame

TL;DR

This paper proves a new downward categoricity transfer in abstract elementary classes with a good frame, improving thresholds for categoricity transfer under tameness and other assumptions.

Contribution

It introduces a novel orthogonality calculus approach to establish downward transfer results in AECs with good frames, extending and refining Shelah's categoricity transfer theorems.

Findings

Downward transfer from categoricity in a successor under a good frame

Improved thresholds for categoricity transfer assuming tameness

Removal of successor hypothesis with additional set-theoretic assumptions

Abstract

We use orthogonality calculus to prove a downward transfer from categoricity in a successor in abstract elementary classes (AECs) that have a good frame (a forking-like notion for types of singletons) on an interval of cardinals: $\mathbf{Theorem}$ Let $K$ be an AEC and let $\text{LS} (K) \le \lambda < \theta$ be cardinals. If $K$ has a type-full good $[\lambda, \theta]$-frame and $K$ is categorical in both $\lambda$ and $\theta^+$, then $K$ is categorical in all $\lambda' \in [\lambda, \theta]$. We deduce improvements on the threshold of several categoricity transfers that do not mention frames. For example, the threshold in Shelah's transfer can be improved from $\beth_{\beth_{\left(2^{\text{LS} (K)}\right)^+}}$ to $\beth_{\left(2^{\text{LS} (K)}\right)^+}$ assuming that the AEC is $\text{LS} (K)$-tame. The successor hypothesis can also be removed from Shelah's result by assuming in addition either that the AEC has primes over sets of the form $M \cup \{a\}$ or (using an unpublished claim of Shelah) that the weak generalized continuum hypothesis holds.