Superstability from categoricity in abstract elementary classes

TL;DR

This paper generalizes a result linking categoricity to superstability-like properties in abstract elementary classes, using a broad notion of independence and weakening amalgamation assumptions, with a proof valid in ZFC.

Contribution

It extends previous results by establishing superstability properties from categoricity under a general independence notion and weaker amalgamation, filling a gap in earlier proofs.

Findings

Superstability-like property derived from categoricity

Generalized independence notion satisfies superstability

Proof valid within ZFC, no additional set-theoretic assumptions

Abstract

Starting from an abstract elementary class with no maximal models, Shelah and Villaveces have shown (assuming instances of diamond) that categoricity implies a superstability-like property for a certain independence relation called nonsplitting. We generalize their result as follows: given an abstract notion of independence for Galois (orbital) types over models, we derive that the notion satisfies a superstability property provided that the class is categorical and satisfies a weakening of amalgamation. This extends the Shelah-Villaveces result (the independence notion there was splitting) as well as a result of the first and second author where the independence notion was coheir. The argument is in ZFC and fills a gap in the Shelah-Villaveces proof.