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arXiv · 1609.07101

Superstability from categoricity in abstract elementary classes

Will Boney·Rami Grossberg·Monica M. VanDieren·Sebastien Vasey
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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.

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Will Boney, Rami Grossberg, Monica M. VanDieren, Sebastien Vasey. 2016-09-22. Superstability from categoricity in abstract elementary classes. https://doi.org/10.1016/j.apal.2017.01.005

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