Tameness from two successive good frames

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 $\lambda$.