A Downward Categoricity Transfer for Tame Abstract Elementary Classes

We prove a downward transfer from categoricity in a successor in tame abstract elementary classes (AECs). This complements the upward transfer of Grossberg and VanDieren and improves the Hanf number in Shelah’s downward transfer (provided the class is tame). Theorem 0.1. Let K be an AEC with amalgamation. If K is LS(K)-weakly tame and categorical in a successor λ ≥ i(2LS(K))+ , then K is categorical in all λ′ ≥ i(2LS(K))+ . The argument uses orthogonality calculus and gives alternate proofs to both the Shelah and the Grossberg-VanDieren transfers. We deduce Shelah’s categoricity conjecture in universal classes with amalgamation: Theorem 0.2. Let K be a universal class with amalgamation and arbitrarily large models. If K is categorical in some λ > |L(K)|+ א0, then K is categorical in all λ′ ≥ min(λ,i(2|L(K)|+א0)+). Heavily using results of Shelah and assuming the weak generalized continuum hypothesis, we can also deal with categoricity in a limit cardinal and prove the categoricity conjecture in tame AECs with amalgamation: Theorem 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 2 + for every cardinal θ, as well as an unpublished claim of Shelah.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 and arbitrarily large models. If K is categorical in some λ > LS(K), then K is categorical in all λ′ ≥ min(λ,i(2LS(K))+). Date: November 9, 2015 AMS 2010 Subject Classification: Primary 03C48. Secondary: 03C45, 03C52, 03C55, 03C75, 03E55.