Tameness from two successive good frames

TL;DR

This paper demonstrates that under certain set-theoretic assumptions, the locality of orbital types in an AEC at a larger cardinality follows from the existence of superstable-like forking notions at two successive cardinalities, linking forking behavior across these sizes.

Contribution

It proves that the locality of orbital types at a larger cardinality can be derived from the existence of good frames at two successive cardinalities, establishing a converse to a known implication.

Findings

Orbital types over models of size λ+ are determined by restrictions to size λ.

Existence of good frames at λ and λ+ implies locality of orbital types at λ+.

Forcing in λ+ can be described in terms of forking in λ.

Abstract

We show, assuming a mild set-theoretic hypothesis, that if an abstract elementary class (AEC) has a superstable-like forking notion for models of cardinality $\lambda$ and a superstable-like forking notion for models of cardinality $\lambda^+$, then orbital types over models of cardinality $\lambda^+$ are determined by their restrictions to submodels of cardinality $\lambda$. By a superstable-like forking notion, we mean here a good frame, a central concept of Shelah's book on AECs. It is known that locality of orbital types together with the existence of a superstable-like notion for models of cardinality $\lambda$ implies the existence of a superstable-like notion for models of cardinality $\lambda^+$, but here we prove the converse. An immediate consequence is that forking in $\lambda^+$ can be described in terms of forking in $\lambda$.