On Prime Models in Totally Categorical Abstract Elementary Classes

We show: Theorem 0.1. Let K be a fully LS(K)-tame and short abstract elementary class (AEC) with amalgamation. Write H1 := i(2LS(K))+ and assume that K is categorical in some λ ≥ H1. The following are equivalent: (1) K≥H1 has primes over sets of the form M ∪ {a}. (2) K is categorical in all λ′ ≥ H1. Note that (1) implies (2) appears in an earlier paper. Here we prove (2) implies (1), generalizing an argument of Shelah who proved the existence of primes at successor cardinals. Assuming a large cardinal axiom, we deduce an equivalence between Shelah’s eventual categoricity conjecture and the statement that every AEC categorical in a proper class of cardinals eventually has prime models over sets of the form M ∪ {a}. Corollary 0.2. Assume there exists a proper class of almost strongly compact cardinals. Let K be an AEC categorical in a proper class of cardinals. The following are equivalent: (1) There exists λ0 such that K≥λ0 has primes over sets of the form M ∪ {a}. (2) There exists λ1 such that K is categorical in all λ ≥ λ1.