Independence in finitary abstract elementary classes

In this paper we study a specific subclass of abstract elementary classes. We construct a notion of independence for these AEC’s and show that under simplicity the notion has all the usual properties of first order non-forking over complete types. Our approach generalizes the context of א0 -stable homogeneous classes and excellent classes. Our set of assumptions follow from disjoint amalgamation, existence of a prime model over ∅ , Löwenheim-Skolem number being ω , LS(K) -tameness and a property we call finite character. We also start the studies of these classes from the א0 -stable case. Stability in א0 and LS(K) -tameness can be replaced by categoricity above the Hanf number. Finite character is the main novelty of this paper. Almost all examples of AEC’s have this property and it allows us to use weak types, as we call them, in place of Galois types.