Rank functions and partial stability spectra for tame AECs

TL;DR

This paper develops rank functions and explores total transcendence in tame abstract elementary classes, establishing connections between stability and transcendence, and proving a significant upward stability transfer result.

Contribution

It introduces new rank functions for Galois types and proves a partial upward stability transfer in tame AECs, advancing classification theory.

Findings

Defined new rank functions for Galois types

Established a connection between stability and total transcendence in tame AECs

Proved a partial upward stability transfer for stable tame AECs

Abstract

We introduce a family of rank functions and related notions of total transcendence for Galois types in abstract elementary classes. We focus, in particular, on abstract elementary classes satisfying the condition know as tameness (currently suspected to be a necessary condition for the development of a reasonable classification theory) where the connections between stability and total transcendence are most evident. As a byproduct, we obtain a partial upward stability transfer result for tame abstract elementary classes stable in a cardinal lambda satisfying lambda^{aleph_0}, a substantial generalization of a result of Baldwin, Kueker, and VanDieren.