Categoricity from one successor cardinal in tame abstract elementary classes. preprint

Abstract. We prove that from categoricity in λ+ we can get categoric-ity in all cardinals ≥ λ+ in a χ-tame abstract elementary classes which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided λ> LS(K) and λ ≥ χ. For the missing case when λ = LS(K), we prove that K is totally categorical provided that K is categorical in LS(K) and LS(K)+. 1. introduction The benchmark of progress in the development of a model theory for abstract elementary classes (AECs) is Shelah’s Categoricity Conjecture. Conjecture 1.1. Let K be an abstract elementary class. If K is categorical in some λ> Hanf(K)1, then for every µ ≥ Hanf(K), K is categorical in µ. With the exception of [MaSh], [KoSh], [Sh 576], [ShVi] and [Va] in which extra set theoretic assumptions are made, all work towards Shelah’s Cat-egoricity Conjecture has taken place under the assumption of the amalga-mation property. An AEC satisfies the amalgamation property if for every triple of models M0,M1,M2 in which M0 ≺K M1 and M0 ≺K M2 there exist K-mappings g1 and g2 and an amalgam N ∈ K such that the diagram below commutes