Shelah's categoricity conjecture from a successor for tame abstract elementary classes

We prove a categoricity transfer theorem for tame abstract elementary classes. Theorem 0.1. Suppose that K is a χ-tame abstract elementary class and satisfies the amalgamation and joint embedding properties and has arbitrarily large models. Let λ ≥ Max{χ,LS(K)}. If K is categorical in λ and λ, then K is categorical in λ. Combining this theorem with some results from [Sh 394], we derive a form of Shelah’s Categoricity Conjecture for tame abstract elementary classes: Corollary 0.2. Suppose K is a χ-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let μ0 := Hanf(K). If χ ≤ i(2μ0 )+ and K is categorical in some λ + > i(2μ0 )+ , then K is categorical in μ for all μ > i(2μ0 )+ .