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

Abstract

We prove a categoricity transfer theorem for tame abstract elementary classes.

Theorem. 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 [37], we derive a form of Shelah’s Categoricity Conjecture for tame abstract elementary classes:

Corollary. Suppose 𝔎 is a χ-tame abstract elementary class satisfying the amalgamation and joint embedding properties. Let μ₀:= Hanf(𝔎). If χ≤ℶ_(2^μ₀)⁺ and 𝔎 is categorical in some λ⁺>ℶ_(2^μ₀)⁺, then 𝔎 is categorical in μ for all μ>ℶ_(2^μ₀)⁺.