Superstability from categoricity in abstract elementary classes
arXiv:1609.07101 · doi:10.1016/j.apal.2017.01.005
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.
14 pages
References in corpus (5)
- Building independence relations in abstract elementary classes
- Shelah's eventual categoricity conjecture in universal classes: part I
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- Downward categoricity from a successor inside a good frame
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Cited by in corpus (6)
- Shelah's eventual categoricity conjecture in universal classes: part I
- Saturation and solvability in abstract elementary classes with amalgamation
- Toward a stability theory of tame abstract elementary classes
- Good Frames in the Hart-Shelah Example
- Shelah's eventual categoricity conjecture in tame AECs with primes
- On categoricity in successive cardinals