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)(<κ)-tame AEC with amalgamation. If  κ=ℶκ>LS(K)κ=ℶκ>LS(K)and K is categorical in a  λ>κλ>κ, then:
• K is stable in any cardinal μ with  μ≥κμ≥κ.
• K is categorical in κ.
• There is a type-full good λ-frame with underlying class  KλKλ.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)λ≥LS(K)such that  K≥λK≥λadmits 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  2λ<2λ+2λ<2λ+for all cardinals λ, as well as an unpublished claim of Shelah.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.