Downward categoricity from a successor inside a good frame

We use orthogonality calculus to prove a downward transfer from categoricity in a successor in abstract elementary classes (AECs) that have a good frame (a forking-like notion for types of singletons) on an interval of cardinals: Theorem 0.1. Let K be an AEC and let LS(K) ≤ λ < θ be cardinals. If K has a type-full good [λ, θ]-frame and K is categorical in both λ and θ, then K is categorical in all λ′ ∈ [λ, θ]. We deduce improvements on the threshold of several categoricity transfers that do not mention frames. For example, we can combine our theorem with the upward transfer of Grossberg and VanDieren to obtain Shelah’s categoricity conjecture from a successor for tame AECs with amalgamation: Corollary 0.2. Let K be a LS(K)-tame AEC with amalgamation. If K is categorical in some successor λ ≥ i(2LS(K))+ , then K is categorical in all λ′ ≥ i(2LS(K))+ . Assuming that the AEC has primes over sets of the form M ∪ {a}, the successor hypothesis can be removed from the previous corollary. It can also be removed by heavily using results of Shelah and assuming the weak generalized continuum hypothesis: Corollary 0.3. Assume 2 < 2 + for every cardinal θ, as well as an unpublished claim of Shelah. Let K be a LS(K)-tame AEC with amalgamation. If K is categorical in some λ ≥ i(2LS(K))+ , then K is categorical in all λ′ ≥ i(2LS(K))+ . Date: April 8, 2016 AMS 2010 Subject Classification: Primary 03C48. Secondary: 03C45, 03C52, 03C55, 03C75, 03E55.