Infinitary stability theory

Abstract

We introduce a new device in the study of abstract elementary classes (AECs): Galois Morleyization, which consists in expanding the models of the class with a relation for every Galois type of length less than a fixed cardinal $\kappa$. We show: $\mathbf{Theorem}$ (The semantic-syntactic correspondence) An AEC $K$ is fully $(<\kappa)$-tame and type short if and only if Galois types are syntactic in the Galois Morleyization. This exhibits a correspondence between AECs and the syntactic framework of stability theory inside a model. We use the correspondence to make progress on the stability theory of tame and type short AECs. The main theorems are: $\mathbf{Theorem}$ Let $K$ be a $\text{LS}(K)$-tame AEC with amalgamation. The following are equivalent: * $K$ is Galois stable in some $\lambda \ge \text{LS}(K)$. * $K$ does not have the order property (defined in terms of Galois types). * There exist cardinals $\mu$ and $\lambda_0$ with $\mu \le \lambda_0 < \beth_{(2^{\text{LS}(K)})^+}$ such that $K$ is Galois stable in any $\lambda \ge \lambda_0$ with $\lambda = \lambda^{<\mu}$. $\mathbf{Theorem}$ Let $K$ be a fully $(<\kappa)$-tame and type short AEC with amalgamation, $\kappa = \beth_{\kappa} > \text{LS} (K)$. If $K$ is Galois stable, then the class of $\kappa$-Galois saturated models of $K$ admits an independence notion ($(<\kappa)$-coheir) which, except perhaps for extension, has the properties of forking in a first-order stable theory.