Downward categoricity from a successor inside a good frame

Abstract

We use orthogonality calculus to prove a downward transfer from categoricity in a successor in abstract elementary classes (AECs) that have a good frame (a forking-like notion for types of singletons) on an interval of cardinals: $\mathbf{Theorem}$ Let $K$ be an AEC and let $\text{LS}(K)\le \lambda <\theta$ be cardinals. If $K$ has a type-full good $[\lambda ,\theta ]$-frame and $K$ is categorical in both $\lambda$ and ${\theta }^{+}$, then $K$ is categorical in all ${\lambda }^{\prime }\in [\lambda ,\theta ]$. We deduce improvements on the threshold of several categoricity transfers that do not mention frames. For example, the threshold in Shelah's transfer can be improved from ${\beth }_{{\beth }_{{\left(2^{\text{LS}(K)}\right)}^{+}}}$ to ${\beth }_{{\left(2^{\text{LS}(K)}\right)}^{+}}$ assuming that the AEC is $\text{LS}(K)$-tame. The successor hypothesis can also be removed from Shelah's result by assuming in addition either that the AEC has primes over sets of the form $M\cup \{a\}$ or (using an unpublished claim of Shelah) that the weak generalized continuum hypothesis holds.

63 pages. Was previously named "A downward categoricity transfer for tame abstract elementary classes"

References in corpus (4)

• Building independence relations in abstract elementary classes
• Infinitary stability theory
• Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes
• Saturation and solvability in abstract elementary classes with amalgamation

Cited by in corpus (11)

• Shelah's eventual categoricity conjecture in universal classes: part I
• Symmetry and the Union of Saturated Models in Superstable Abstract Elementary Classes
• Saturation and solvability in abstract elementary classes with amalgamation
• Superstability from categoricity in abstract elementary classes
• Universal classes near ${\aleph }_1$
• Abstract elementary classes stable in ${\aleph }_0$
• Building prime models in fully good abstract elementary classes
• Good Frames in the Hart-Shelah Example
• Shelah's eventual categoricity conjecture in tame AECs with primes
• Tameness from two successive good frames
• Density of uniqueness triples from the diamond axiom