Building independence relations in abstract elementary classes

Abstract

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 $(<\kappa )$-tame AEC with amalgamation. If $\kappa ={\beth }_{\kappa }>\text{LS}(K)$ and $K$ is categorical in a $\lambda >\kappa$, then: * $K$ is stable in all cardinals $\ge \kappa$. * $K$ 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). $\mathbf{Theorem}$ (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 $\lambda \ge \text{LS}(K)$ such that $K_{\ge \lambda }$ 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. $\text{Corollary}$ 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2^{{\lambda }^{+}}$ for all cardinals $\lambda$, as well as an unpublished claim of Shelah.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.

96 pages. Was initially part of Infinitary stability theory (arXiv:1412.3313). Early versions were called "Independence in abstract elementary classes"

References in corpus (3)

• Infinitary stability theory
• Chains of saturated models in AECs
• Building prime models in fully good abstract elementary classes

Cited by in corpus (20)

• Shelah's eventual categoricity conjecture in universal classes: part I
• Forking independence from the categorical point of view
• Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes
• Shelah's eventual categoricity conjecture in universal classes. Part II
• Saturation and solvability in abstract elementary classes with amalgamation
• Chains of saturated models in AECs
• Downward categoricity from a successor inside a good frame
• Superstability from categoricity in abstract elementary classes
• Toward a stability theory of tame abstract elementary classes
• Symmetry in abstract elementary classes with amalgamation
• Building prime models in fully good abstract elementary classes
• Abstract elementary classes stable in ${\aleph }_0$
• Good Frames in the Hart-Shelah Example
• Shelah's eventual categoricity conjecture in tame AECs with primes
• Building models in small cardinals in local abstract elementary classes
• NSOP${}_1$-like independence in AECats
• Tameness from two successive good frames
• Non-forking w-good frames
• An NIP-like Notion in Abstract Elementary Classes
• Lifting independence along functors