Saturation and solvability in abstract elementary classes with amalgamation

Theorem 0.1. Let K be an abstract elementary class (AEC) with amalgamation and no maximal models. Let λ > LS(K). If K is categorical in λ, then the model of cardinality λ 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 λ, (when λ is big-enough) K is weakly tame below λ, 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): Corollary 0.2. Let K be an AEC with amalgamation and no maximal models. Let λ > μ > LS(K). If K is solvable in λ, then K is solvable in μ.