Saturation and solvability in abstract elementary classes with amalgamation

This is the title of a talk given at the 2017 North American meeting of the ASL at Boise State University on March 21, 2017. The talk presents the corresponding paper. Here are the slides.


Abstract elementary classes (AECs) are an axiomatic framework encompassing classes of models of an L∞, ω theory, as well as numerous algebraic examples. They were introduced by Saharon Shelah forty years ago. Shelah focused on generalizations of Morley's categoricity theorem and conjectured the following eventual version: An AEC categorical in a high-enough cardinal is categorical on a tail of cardinals.

In a milestone paper, Shelah proved a downward version of the conjecture for AECs with amalgamation when the categoricity cardinal is a successor. In this talk, we will deal with AECs with amalgamation categorical in a not necessarily successor cardinal. We will discuss the generalization of the first step of the proof of Morley's categoricity theorem to that setup:

Theorem. Let K$ be an AEC with amalgamation and let λ > LS(K). If K is categorical in λ, then the model of cardinality λ is saturated (for orbital types).

The proof uses several results from the superstability theory of AECs, including criterias for the uniqueness of limit models. Time permitting, we will also talk about solvability, a weak form of categoricity introduced by Shelah in his book on AECs as a generalization of superstability. An application of the proof of the above theorem is a downward transfer of solvability:

Theorem. Let K be an AEC with amalgamation and let λ > μ > LS(K). If K is solvable in λ, then K is solvable in μ.