Saturation and solvability in abstract elementary classes with amalgamation

Abstract

$\mathbf{Theorem.}$ Let $K$ be an abstract elementary class (AEC) with amalgamation and no maximal models. Let $\lambda > \text{LS} (K)$. If $K$ is categorical in $\lambda$, then the model of cardinality $\lambda$ is Galois-saturated. This answers a question asked independently by Baldwin and Shelah. We deduce several corollaries: $K$ has a unique limit model in each cardinal below $\lambda$, (when $\lambda$ is big-enough) $K$ is weakly tame below $\lambda$, and the thresholds of several existing categoricity transfers can be improved. We also prove a downward transfer of solvability (a version of superstability introduced by Shelah): $\mathbf{Corollary.}$ Let $K$ be an AEC with amalgamation and no maximal models. Let $\lambda > \mu > \text{LS} (K)$. If $K$ is solvable in $\lambda$, then $K$ is solvable in $\mu$.