Toward a stability theory of tame abstract elementary classes

TL;DR

This paper develops a stability theory for tame abstract elementary classes, characterizing stability cardinals and linking stability with saturated models, with implications for homogeneous model theory.

Contribution

It provides a full characterization of stability cardinals under SCH and connects stability spectrum with saturated models, advancing the understanding of tame AECs.

Findings

Characterization of stability cardinals assuming SCH

Stable classes have no long splitting chains

Stability in a tail of cardinals implies broader stability in homogeneous diagrams

Abstract

We initiate a systematic investigation of the abstract elementary classes that have amalgamation, satisfy tameness (a locality property for orbital types), and are stable (in terms of the number of orbital types) in some cardinal. Assuming the singular cardinal hypothesis (SCH), we prove a full characterization of the (high-enough) stability cardinals, and connect the stability spectrum with the behavior of saturated models. We deduce (in ZFC) that if a class is stable on a tail of cardinals, then it has no long splitting chains (the converse is known). This indicates that there is a clear notion of superstability in this framework. We also present an application to homogeneous model theory: for $D$ a homogeneous diagram in a first-order theory $T$, if $D$ is both stable in $|T|$ and categorical in $|T|$ then $D$ is stable in all $\lambda \ge |T|$.