Disjoint non-forking amalgamation in stable AECs

TL;DR

This paper proves that in certain stable abstract elementary classes, high cofinality limit models serve as disjoint non-forking amalgamation bases, advancing understanding of amalgamation properties in model theory.

Contribution

It establishes disjoint non-forking amalgamation for high cofinality limit models in stable AECs with specific independence relations, extending prior results on amalgamation.

Findings

High cofinality limit models are disjoint non-forking amalgamation bases.

Under certain stability and independence conditions, disjoint amalgamation is guaranteed.

Results connect amalgamation properties with stability and independence in AECs.

Abstract

The disjoint amalgamation property (DAP), which asserts that all spans of a class of models can be amalgamated with minimal intersection, is an important property in the context of abstract elementary classes, with connections to both Grossberg's question and Shelah's categoricity conjecture. We prove that, in a nice AEC $\mathbf{K}$ stable in $\lambda \geq \operatorname{LS}(\mathbf{K})$ with a strong enough independence relation, all high cofinality $\lambda$-limit models are disjoint (non-forking) amalgamation bases. $\textbf{Theorem.}$ Let $\mathbf{K}$ be an AEC stable in $\lambda$, where $\mathbf{K}_\lambda$ has AP, JEP, and NMM, and let $\mathbf{K}'$ be some AC where $\mathbf{K}_{(\lambda,\geq\kappa)} \subseteq \mathbf{K}' \subseteq \mathbf{K}_\lambda$. Suppose there is an independence relation on $\mathbf{K}'$ satisfying uniqueness, existence, non-forking amalgamation, $\mathbf{K}_{(\lambda,\geq\kappa)}$-universal continuity* in $\mathbf{K}_\lambda$, and $(\geq \kappa)$-local character. Assume $M_0, M_1, M_2 \in \mathbf{K}_{(\lambda,\geq\kappa)}$, and that $M_0 \leq_{\mathbf{K}} M_l$ and $a_l \in M_l$ for $l = 1, 2$. Then there exist $N \in \mathbf{K}_{(\lambda,\geq\kappa)}$ and $f_l : M_l \rightarrow N$ fixing $M_0$ for $l = 1, 2$ such that $\operatorname{gtp}(f_l(a_l)/f_{3-l}[M_{3-l}], N)$ does not fork over $M_0$ and $f_1[M_1] \cap f_2[M_2] = M_0$. That is, our independence relation has disjoint non-forking amalgamation in $\mathbf{K}_{(\lambda,\geq\kappa)}$. In particular, every $M_0 \in \mathbf{K}_{(\lambda,\geq\kappa)}$ is a disjoint amalgamation base in $\mathbf{K}_\lambda$. The hypotheses on the independence relation can be weakened (closer to $\lambda$-non-splitting in $\lambda$-stable AECs) if we are willing to give up the `non-forking' conditions of the amalgamation.