Forking and superstability in tame AECs

TL;DR

This paper establishes the existence of a forking-like independence notion in tame AECs under categoricity or superstability-like assumptions, advancing the understanding of stability and model uniqueness.

Contribution

It provides the first general construction of a good frame in ZFC for tame AECs and introduces new conditions for limit model uniqueness.

Findings

Existence of an eventually global good frame in tame AECs under categoricity.

Independence relations derived from superstability-like hypotheses.

Upward stability transfer and new criteria for limit model uniqueness.

Abstract

We prove that any tame abstract elementary class categorical in a suitable cardinal has an eventually global good frame: a forking-like notion defined on all types of single elements. This gives the first known general construction of a good frame in ZFC. We show that we already obtain a well-behaved independence relation assuming only a superstability-like hypothesis instead of categoricity. These methods are applied to obtain an upward stability transfer theorem from categoricity and tameness, as well as new conditions for uniqueness of limit models.