Uniqueness of limit models in classes with amalgamation

We prove: Theorem 0.1 (Main Theorem). Let K be an AEC and μ > LS(K). Suppose K satisfies the disjoint amalgamation property for models of cardinality μ. If K is μ-Galois-stable, does not have long splitting chains, and satisfies locality of splitting, then any two (μ, σl)-limits over M for (l ∈ {1, 2}) are isomorphic over M . This result extends results of Shelah from [Sh 394], [Sh 576], [Sh 600], Kolman and Shelah in [KoSh] and Shelah & Villaveces from [ShVi]. Our uniqueness theorem was used by Grossberg and VanDieren to prove a case of Shelah’s categoricity conjecture for tame AEC in [GrVa2].