On the problem of stability of abstract elementary classes of modules
Gianluca Paolini, Saharon Shelah
TL;DR
This work answers Mazari-Armida's open question by constructing an unstable abstract elementary class of torsion-free abelian groups under pure submodule inclusion, showing the stability question for such AECs can fail. It also develops a robust stability framework for AECs of R-modules, proving κ-local (κ-tame) AECs are almost stable in ZFC and that amalgamation elevates almost stability to stability; with a strongly compact cardinal, these AECs become stable. The authors introduce and leverage the notion of almost stability, clarifying its equivalence to stability under amalgamation and tying the results to the first-order theory of R-modules. Finally, they provide a concrete counterexample illustrating the boundary between stability and instability in module-based AECs, highlighting the nuanced landscape of AEC stability beyond the first-order setting.
Abstract
It is an open problem of Mazari-Armida whether every abstract elementary class of $R$-modules $(\mathbf{K}, \leq_{\mathrm{pure}})$, with $\leq_{\mathrm{pure}}$ the pure submodule relation, is stable. We answer this question in the negative by constructing unstable abstract elementary classes $(\mathbf{K}, \leq_{\mathrm{pure}})$ of torsion-free abelian groups. On the other hand, we prove (in $\mathrm{ZFC}$) that if $R$ is any ring and $(\mathbf{K}, \preccurlyeq)$ is an abstract elementary class of $R$-modules which is $κ$-local (also called $κ$-tame) for some $κ\geq \mathrm{LS}(\mathbf{K}, \preccurlyeq)$, then $(\mathbf{K}, \preccurlyeq)$ is almost stable, where almost stability is a new notion of independent interest that we introduce in this paper, and which is equivalent to the usual notion of stability under the assumption of amalgamation. As a consequence, assuming the existence of a strongly compact cardinal $κ$, we have that every abstract elementary class $(\mathbf{K}, \preccurlyeq)$ of $R$-modules with amalgamation satisfying $κ> \mathrm{LS}(\mathbf{K}, \preccurlyeq)$ is stable.