Superstability in simple finitary AECs
Volume 195 / 2007
Abstract
We continue the study of {finitary abstract elementary classes} beyond $\aleph_0$-stability. We suggest a possible notion of superstability for simple finitary AECs, and derive from this notion several good properties for independence. We also study constructible models and the behaviour of Galois types and {weak Lascar strong types} in this context. We show that superstability is implied by {a-categoricity} in a suitable cardinal. As an application we prove the following theorem: Assume that $(\mathbb{K},\preccurlyeq_\mathbb{K})$ is a simple, tame, finitary AEC, a-categorical in some cardinal $\kappa$ above the Hanf number such that $\mathop{\rm cf}\nolimits(\kappa)>\omega$. Then $(\mathbb{K},\preccurlyeq_\mathbb{K})$ is a-categorical in each cardinal above the Hanf number.
