Forking and superstability in tame AECs

Abstract

We prove that any tame abstract elementary class categorical in a suitable cardinal has an eventually global good frame: a forking-like notion defined on all types of single elements. This gives the first known general construction of a good frame in ZFC. We show that we already obtain a well-behaved independence relation assuming only a superstability-like hypothesis instead of categoricity. These methods are applied to obtain an upward stability transfer theorem from categoricity and tameness, as well as new conditions for uniqueness of limit models.

33 pages

Cited by in corpus (17)

• Canonical forking in AECs
• Building independence relations in abstract elementary classes
• Shelah's eventual categoricity conjecture in universal classes: part I
• $\mu$-Abstract Elementary Classes and other generalizations
• Chains of saturated models in AECs
• Saturation and solvability in abstract elementary classes with amalgamation
• Tameness and frames revisited
• Superstability from categoricity in abstract elementary classes
• Downward categoricity from a successor inside a good frame
• 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
• Shelah's eventual categoricity conjecture in tame AECs with primes
• Good Frames in the Hart-Shelah Example
• Building models in small cardinals in local abstract elementary classes
• On categoricity in successive cardinals
• An NIP-like Notion in Abstract Elementary Classes