Tameness from two successive good frames

Abstract

We show, assuming a mild set-theoretic hypothesis, that if an abstract elementary class (AEC) has a superstable-like forking notion for models of cardinality $\lambda$ and a superstable-like forking notion for models of cardinality ${\lambda }^{+}$, then orbital types over models of cardinality ${\lambda }^{+}$ are determined by their restrictions to submodels of cardinality $\lambda$. By a superstable-like forking notion, we mean here a good frame, a central concept of Shelah's book on AECs. It is known that locality of orbital types together with the existence of a superstable-like notion for models of cardinality $\lambda$ implies the existence of a superstable-like notion for models of cardinality ${\lambda }^{+}$, but here we prove the converse. An immediate consequence is that forking in ${\lambda }^{+}$ can be described in terms of forking in $\lambda$.

27 pages

References in corpus (9)

• Building independence relations in abstract elementary classes
• Shelah's eventual categoricity conjecture in universal classes: part I
• Infinitary stability theory
• 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
• Downward categoricity from a successor inside a good frame
• Symmetry in abstract elementary classes with amalgamation
• Good Frames in the Hart-Shelah Example

Cited by in corpus (1)

• On categoricity in successive cardinals