Submission date: 26 April 2016

Abstract:

Theorem.
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.
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.

Mathematics Subject Classification: 03C48 (Primary), 03C45, 03C52, 03C55 (Secondary)

Keywords and phrases:

Full text arXiv 1604.07743: pdf, ps.