Tameness and Extending Frames

TL;DR

This paper demonstrates that combining tameness and good λ-frames in Abstract Elementary Classes (AECs) results in a well-behaved nonforking notion across all cardinalities, advancing classification theory.

Contribution

It shows that tameness and good λ-frames together produce a robust nonforking framework in AECs, filling a key gap in the field.

Findings

Established a well-behaved nonforking notion in all cardinalities.

Proved a complete stability transfer theorem.

Demonstrated uniqueness of limit models in tame AECs.

Abstract

We combine two notions in AECs, tameness and good $\lambda$-frames, and show that they together give a very well-behaved nonforking notion in all cardinalities. This helps to fill a longstanding gap in classification theory of tame AECs and increases the applicability of frames. Along the way, we prove a complete stability transfer theorem and uniqueness of limit models in these AECs.