Saturation and solvability in abstract elementary classes with amalgamation

Abstract

$Theorem.$ 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): $Corollary.$ 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 $\mu$.

19 pages

References in corpus (9)

• Building independence relations in abstract elementary classes
• Shelah's eventual categoricity conjecture in universal classes: part I
• Infinitary stability theory
• Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes
• Shelah's eventual categoricity conjecture in universal classes. Part II
• Downward categoricity from a successor inside a good frame
• Superstability from categoricity in abstract elementary classes
• Chains of saturated models in AECs
• Symmetry in abstract elementary classes with amalgamation

Cited by in corpus (7)

• Superstability from categoricity in abstract elementary classes
• Downward categoricity from a successor inside a good frame
• Toward a stability theory of tame abstract elementary classes
• On categoricity in successive cardinals
• Tameness from two successive good frames
• Accessible categories, set theory, and model theory: an invitation
• Shelah-Villaveces revisited