Paper detail

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. $\mathbf{Theorem}$ (Superstability from categoricity) Let $K$ be a $(<κ)$-tame AEC with amalgamation. If $κ= \beth_κ> \text{LS} (K)$ and $K$ is categorical in a $λ> κ$, then: * $K$ is stable in all cardinals $\ge κ$. * $K$ is categorical in $κ$. * There is a type-full good $λ$-frame with underlying class $K_λ$. Under more locality conditions, we prove that the frame extends to a global independence notion (for types of arbitrary length). $\mathbf{Theorem}$ (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 $λ\ge \text{LS} (K)$ such that $K_{\ge λ}$ 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. $\textbf{Corollary}$ Assume $2^λ < 2^{λ^+}$ for all cardinals $λ$, as well as an unpublished claim of Shelah.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.