Sebastien Vasey

FORKING AND SUPERSTABILITY IN TAME AECS

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.

Canonical forking in AECs

Boney and Grossberg (BG) proved that every nice AEC has an in- dependence relation. We prove that this relation is unique: In any given AEC, there can exist at most one independence relation that satisfies existence, exten- sion, uniqueness and local character. While doing this, we study more generally properties of independence relations for AECs and also prove a canonicity re- sult for Shelahs good frames. The usual tools of first-order logic (like the finite equivalence relation theorem or the type amalgamation theorem in simple theo- ries) are not available in this context. In addition to the loss of the compactness theorem, we have the added difficulty of not being able to assume that types are sets of formulas. We work axiomatically and develop new tools to understand this general framework.

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. Theorem 0.1 Superstability from categoricity Let K be a ( κ ) -tame AEC with amalgamation. If κ = ℶ κ > LS ( K ) and K is categorical in a λ > κ , then: • K is stable in any cardinal μ with μ ≥ 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). Theorem 0.2 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 ≥ LS ( K ) such that K ≥ 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 Shelahs eventual categoricity conjecture for AECs (without assuming categoricity in a successor cardinal) follows from the weak generalized continuum hypothesis and a large cardinal axiom. Corollary 0.3 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.

Chains of saturated models in AECs

AbstractWe study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove: Theorem 0.1.IfKis a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough

TAMENESS AND FRAMES REVISITED

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Saturation and solvability in abstract elementary classes with amalgamation

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EQUIVALENT DEFINITIONS OF SUPERSTABILITY IN TAME ABSTRACT ELEMENTARY CLASSES

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