Paper

Disjoint non-forking amalgamation in stable AECs

Beard, Jeremy

2026

Overview

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 \\(K\\) stable in \\( LS(K)\\) with a strong enough independence relation, all high cofinality \\(\\)-limit models are disjoint (non-forking) amalgamation bases. \\(Theorem.\\) Let \\(K\\) be an AEC stable in \\(\\), where \\(K_\\) has AP, JEP, and NMM, and let \\(K'\\) be some AC where \\(K_( ) K' K_\\). Suppose there is an independence relation on \\(K'\\) satisfying uniqueness, existence, non-forking amalgamation, \\(K_( )\\)-universal continuity* in \\(K_\\), and \\(( )\\)-local character. Assume \\(M_0, M_1, M_2 ın K_( )\\), and that \\(M_0 _K M_l\\) and \\(a_l ın M_l\\) for \\(l = 1, 2\\). Then there exist \\(N ın K_( )\\) and \\(f_l : M_l N\\) fixing \\(M_0\\) for \\(l = 1, 2\\) such that \\(gtp(f_l(a_l)/f_3-l[M_3-l], N)\\) does not fork over \\(M_0\\) and \\(f_1[M_1] f_2[M_2] = M_0\\). That is, our independence relation has disjoint non-forking amalgamation in \\(K_( )\\). In particular, every \\(M_0 ın K_( )\\) is a disjoint amalgamation base in \\(K_\\). The hypotheses on the independence relation can be weakened (closer to \\(\\)-non-splitting in \\(\\)-stable AECs) if we are willing to give up the `non-forking' conditions of the amalgamation.

• Publisher: Cornell University Library, arXiv.org
• Subject: Amalgamation