Abstract
In the setting of abstract elementary classes (AECs) with amalgamation, Shelah has proven a downward categoricity transfer from categoricity in a successor and Grossberg and VanDieren have established an upward transfer assuming in addition a locality property for Galois types that they called tameness.We further investigate categoricity transfers in tame AECs. We use orthogonality calculus to prove a downward transfer from categoricity in a successor in AECs that have a good frame (a forking-like notion for types of singletons) on an interval of cardinals:Theorem 0.1LetKbe an AEC and letLS(K)≤λ<θbe cardinals. IfKhas a type-full good[λ,θ]-frame andKis categorical in both λ andθ+, thenKis categorical in allμ∈[λ,θ].We deduce improvements on the threshold of several categoricity transfers that do not mention frames. For example, the threshold in Shelah's transfer can be improved from ℶℶ(2LS(K))+ to ℶ(2LS(K))+ assuming that the AEC is LS(K)-tame. The successor hypothesis can also be removed from Shelah's result by assuming in addition either that the AEC has primes over sets of the form M∪{a} or (using an unpublished claim of Shelah) that the weak generalized continuum hypothesis holds.
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