Building independence relations in abstract elementary classes

作者：Vasey Sebastien^*

来源：Annals of Pure and Applied Logic, 2016, 167(11): 1029-1092.

DOI：10.1016/j.apal.2016.04.005

• Abstract elementary classes
• Forking
• Good frames
• Categoricity
• Superstability
• Tameness

We study general methods to build forking-like notions in the framework of tame abstract elementary classes (AECs) with amalgamation. We show that whenever such classes are categorical in a high-enough cardinal, they admit a good frame: a forking-like notion for types of singleton elements.
Theorem 0.1 (Superstability from categoricity). Let K be a (< kappa)-tame AEC with amalgamation. If kappa = beth(kappa) > LS(K) and K is categorical in a lambda > kappa, then:
K is stable in any cardinal mu with mu >= kappa.
K is categorical in kappa.
There is a type-full good lambda-frame with underlying class K-lambda.
Under more locality conditions, we prove that the frame extends to a global independence notion (for types of arbitrary length).
Theorem 0.2 (A global independence notion from categoricity). Let K be a densely type-local, fully tame and type short AEC with amalgamation. If K is categorical in unboundedly many cardinals, then there exists lambda >= LS(K) such that K->=lambda), admits a global independence relation with the properties of forking in a superstable first order theory.
As an application, we deduce (modulo an unproven claim of Shelah) that Shelah's eventual categoricity conjecture for AECs (without assuming categoricity in a successor cardinal) follows from the weak generalized continuum hypothesis and a large cardinal axiom.
Corollary 0.3. Assume 2(lambda) < 2(lambda+) for all cardinals lambda, as well as an unpublished claim of Shelah. If there exists a proper class of strongly compact cardinals, then any AEC categorical in some high-enough cardinal is categorical in all high-enough cardinals.

• 出版日期2016-11