Submission date: 27 July 2017

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 λ and a superstable-like forking notion for models of cardinality λ^+, then orbital types over saturated 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^+, but here we prove the converse. An immediate consequence is that forking in λ^+ can be described in terms of forking in λ.

Mathematics Subject Classification: 03C48 (Primary), 03C45, 03C52, 03C55, 03C75 (Secondary)

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Full text arXiv 1707.09008: pdf, ps.