Local Character Expansion in AECs

• Local Character Expansion is a method ensuring that any type over a large model stabilizes over a small submodel, reflecting full local character.
• It employs sequences of models and long transitivity to construct a non-forking relation that meets all good frame axioms in AECs.
• This enhanced framework constrains the spectrum function and supports advanced stability and categoricity analyses in abstract elementary classes.

A local character expansion is a fundamental tool in stability theory and classification theory for abstract elementary classes (AECs). It refers to a property of a non-forking independence relation ensuring that the behavior of types in a large model stabilizes over some small submodel, mirroring properties from superstable first order theories. In the context of AECs with a non-forking relation, the local character asserts that for any type over a sufficiently large model, there exists a “small” submodel over which the type does not fork, providing control over the complexity of types and the spectrum of model cardinalities.

1. Enhancement of Non-Forking Relations and Local Character

The original non-forking relation for AECs—defined in [Sh E46]—mimics basic properties of non-forking in first order stable theories but only achieves a weak version of local character. The current work develops a strengthened non-forking relation, denoted nfNF, that satisfies the full local character property, as well as the other axioms of a good A-frame: monotonicity, existence, uniqueness, symmetry, and continuity. The improved local character axiom (Definition 2.1(c)) stipulates that for every limit ordinal $\delta < \lambda^+$, if $(M_i : i \leq \delta)$ is an increasing continuous sequence in $K_\lambda$ and $$\mathrm{ga-tp}(a, M_\delta, N) \in S^{\mathrm{bs}}(M_\delta)$$, then there is $i < \delta$ such that $\mathrm{ga-tp}(a, M_\delta, N)$ does not fork over $M_i$.

This strengthening is achieved by constructing sequences of models and applying long transitivity of the core NF relation, combined with auxiliary constructions (Definition 4.2), and is proven to satisfy all good frame axioms in Theorem 4.3.

2. Local Character in Classification Theory

The local character property stipulates the existence, for every type $p$ over an increasing union $M_\delta$ of a continuous chain $(M_i)_{i\leq\delta}$, of some $i<\delta$ so that $p$ does not fork over $M_i$. This is a structural constraint ensuring that:

• Types are controlled by small submodels: Complexity cannot “accumulate” indefinitely; rather, the behavior of a type stabilizes at some stage in the chain.
• Stability and spectrum control: The existence of small “bases” for types is crucial for deducing stability properties and for limiting the proliferation of nonisomorphic models, thereby constraining the spectrum function $\lambda \mapsto I(\lambda,K)$.

In superstable first order theories, this property is essential for developing a fine-grained analysis of forking, regular types, and dimensions; the extension of local character to AECs enables similarly powerful control.

3. Model-Theoretic Formulation

The enhanced local character is encoded as an axiom (Definition 2.1(c)):

$$\forall (M_i : i \leq \delta),\ M_i \in K_\lambda \text{ increasing and continuous},\ \delta < \lambda^+ \text{ limit},$$

$$\left[ \mathrm{ga-tp}(a, M_\delta, N) \in S^{\mathrm{bs}}(M_\delta) \implies \exists i < \delta: \mathrm{ga-tp}(a, M_\delta, N)\ \text{does not fork over}\ M_i \right].$$

To guarantee this, the induction constructs an auxiliary chain $(N_{a,i})_{i \leq \delta+1}$ satisfying, for each $i < \delta$, the non-forking relation:

$$\mathrm{NF}(N_{a,i}, N_{a,i+1}, N_{a+1,i}, N_{a+1,i+1}).$$

By long transitivity (see proof of Theorem 4.3), this sequence guarantees non-forking of the eventual type over one of the $M_i$, enforcing local character.

4. Impact on Spectrum and Categoricity

A critical consequence of the improved local character is the resulting constraints on the spectrum function $\lambda \mapsto I(\lambda,K)$, where $I(\lambda,K)$ is the number of models (up to isomorphism) of cardinality $\lambda$. As shown in Section 5, under categoricity and the full set of good frame axioms:

• Arbitrary behaviors for the spectrum function are prohibited: Certain configurations—for example, $$I(\lambda) = I(\lambda^+) = \cdots = I(\lambda^{+(n-1)})=1$$ but $I(\lambda^{+n})=0$ for some $n \ge 3$—are shown to be impossible.
• Fine control over the number and structure of models: The local character ensures that stability-theoretic techniques can be imported to AECs, preventing wild combinatorial growth at higher cardinalities.

Explicitly, the assumed constraints, together with local character and other frame properties, yield contradictions unless the spectrum function behaves in a tightly controlled fashion.

5. Comparison with Prior Results and Technical Advances

Compared to [Sh E46], which produced a weak form of local character, the present results achieve:

• Full realization of all good frame axioms, including local character (Theorem 4.3),
• The introduction of the nfNF relation, defined via long transitivity and conjugation (Definitions 3.2–3.4, Theorem 3.8),
• A robust, “full” good A-frame—enabling powerful model-theoretic conclusions that closely mirror those possible in superstable first order logic.

As a result, the development bridges a crucial gap between first order and non-elementary classification—the improved framework now supports advanced categoricity and stability analyses.

6. Technical Summary and Consequences

The refinements to the non-forking framework for AECs yield:

• A relation satisfying local character: As formalized via sequences of models, non-forking, and the use of auxiliary chains, this ensures types always “base” over proper subchains in continuous sequences.
• A spectrum of models that is not arbitrary: The behavior $\lambda \mapsto I(\lambda,K)$ is governed by the frame axioms, translating local properties of types into global constraints on AEC structure (Fact 5.1, Theorem 5.2).
• Enhanced axiomatic coverage: All desired properties—monotonicity, uniqueness, symmetry, existence, continuity, and now local character—are satisfied, fully generalizing the non-forking framework to the superstable context for AECs.

This integrated system facilitates a much closer parallel to superstable first order theory, solidifying the foundation for further advances in classification in non-elementary model theory.