Building independence relations in abstract elementary classes

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 (< κ)-tame AEC with amalgamation. If κ = iκ > LS(K) and K is categorical in a λ > κ, then: • K is stable in any cardinal μ with μ ≥ κ. • K is categorical in κ. • There is a type-full good λ-frame with underlying class Kλ. 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 λ ≥ LS(K) such that K≥λ 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 < 2 + for all cardinals λ, as well as an unpublished claim of Shelah.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. Date: March 29, 2016 AMS 2010 Subject Classification: Primary 03C48. Secondary: 03C45, 03C52, 03C55, 03C75, 03E55.