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Infinitary stability theory
Sebastien Vasey
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We introduce a new device in the study of abstract elementary classes (AECs): Galois Morleyization, which consists in expanding the models of the class with a relation for every Galois type of length less than a fixed cardinal κ. We show: 𝐓𝐡𝐞𝐨𝐫𝐞𝐦 (The semantic-syntactic correspondence) An AEC K is fully (<κ)-tame and type short if and only if Galois types are syntactic in the Galois Morleyization. This exhibits a correspondence between AECs and the syntactic framework of stability theory inside a model. We use the correspondence to make progress on the stability theory of tame and type short AECs. The main theorems are: 𝐓𝐡𝐞𝐨𝐫𝐞𝐦 Let K be a LS(K)-tame AEC with amalgamation. The following are equivalent: * K is Galois stable in some λ≥LS(K). * K does not have the order property (defined in terms of Galois types). * There exist cardinals μ and λ₀ with μ≤λ₀ < ℶ_((2^(LS(K)))^+) such that K is Galois stable in any λ≥λ₀ with λ= λ^(<μ). 𝐓𝐡𝐞𝐨𝐫𝐞𝐦 Let K be a fully (<κ)-tame and type short AEC with amalgamation, κ= ℶ_κ > LS (K). If K is Galois stable, then the class of κ-Galois saturated models of K admits an independence notion ((<κ)-coheir) which, except perhaps for extension, has the properties of forking in a first-order stable theory.
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