Disjoint non-forking amalgamation in stable AECs

TL;DR

The paper addresses disjoint amalgamation in stable AECs by proving that, under a robust independence framework with uniqueness, existence, non-forking amalgamation, universal continuity*, and $(oldsymbol{\ge}oldsymbol{\kappa})$-local character, high cofinality $oldsymbol{\lambda}$-limit models are disjoint non-forking amalgamation bases. The authors advance a tower-based method, constructing reduced towers and employing tower amalgamation to realize a matrix where the images of two extensions intersect exactly in the base, while preserving non-forking of singleton elements. This yields a disjoint non-forking amalgamation result, implying every base $M_0$ in $ extbf{K}_{(oldsymbol{\lambda}, oldsymbol{\ge}oldsymbol{\kappa})}$ is a disjoint amalgamation base in $ extbf{K}_oldsymbol{\lambda}$, and extends prior work (notably vas19) by reducing categoricity assumptions and broadening applicability to strictly stable AECs. The paper also explores weaker independence notions, showing a similar disjoint amalgamation outcome for high cofinality limit models, though with the caveat that non-forking for singleton elements may not hold, leaving an open question about weak disjoint non-forking amalgamation. Overall, the results connect disjoint amalgamation to independence theory in stable AECs, contributing to Grossberg’s and Shelah’s program by clarifying when disjointness can be achieved via non-forking amalgamation in a non-first-order context.

Abstract

The \emph{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 $λ\geq \operatorname{LS}(\mathbf{K})$ with a strong enough independence relation, all high cofinality $λ$-limit models are disjoint (non-forking) amalgamation bases. $\textbf{Theorem.}$ Let $\mathbf{K}$ be an AEC stable in $λ$, where $\mathbf{K}_λ$ has AP, JEP, and NMM, and let $\mathbf{K}'$ be some AC where $\mathbf{K}_{(λ,\geqκ)} \subseteq \mathbf{K}' \subseteq \mathbf{K}_λ$. Suppose there is an independence relation on $\mathbf{K}'$ satisfying uniqueness, existence, non-forking amalgamation, $\mathbf{K}_{(λ,\geqκ)}$-universal continuity* in $\mathbf{K}_λ$, and $(\geq κ)$-local character. Assume $M_0, M_1, M_2 \in \mathbf{K}_{(λ,\geqκ)}$, and that $M_0 \leq_{\mathbf{K}} M_l$ and $a_l \in M_l$ for $l = 1, 2$. Then there exist $N \in \mathbf{K}_{(λ,\geqκ)}$ 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 particular, every $M_0 \in \mathbf{K}_{(λ,\geqκ)}$ is a disjoint amalgamation base in $\mathbf{K}_λ$. The hypotheses on the independence relation can be weakened (closer to $λ$-non-splitting in $λ$-stable AECs) if we are willing to give up the `non-forking' conditions of the amalgamation.

Figures (5)

• Figure 1:
• Figure 2: The main construction
• Figure 3: The construction from Lemma \ref{['type-ap-lim-isomorphism-lemma']}.
• Figure 4: The tower amalgamation construction in Proposition \ref{['tower-ap']}.
• Figure 5: $(\lambda, \theta)$-weak non-forking amalgamation says that, given the solid black diagram, the dotted blue diagram exists.

Theorems & Definitions (75)

• Definition 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Example 2.4
• Definition 2.5
• Definition 2.8
• Definition 2.9
• Definition 2.10