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Preprint Number 842

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842. Sebastien Vasey
Independence in abstract elementary classes
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Submission date: 4 March 2015.

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 (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 all cardinals \ge 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 (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 \ge LS(K) such that K_{\ge λ} admits a global independence relation with the properties of forking in a superstable first-order theory.
Modulo an unproven claim of Shelah, we deduce that Shelah's categoricity conjecture follows from the weak generalized continuum hypothesis and the existence of unboundedly many strongly compact cardinals.

Mathematics Subject Classification: 03C48 (Primary), 03C45, 03C52, 03C55, 03C75, 03E55 (Secondary)

Keywords and phrases:

Full text arXiv 1503.01366: pdf, ps.


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