Tameness from Two Successive Good Frames

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 λ and a superstable-like forking notion for models of cardinality λ+, then orbital types over models of cardinality λ+ are determined by their restrictions to submodels of cardinality λ. 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 λ implies the existence of a superstable-like notion for models of cardinality λ+, and here we prove the converse. An immediate consequence is that forking in λ+ can be described in terms of forking in λ.