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arXiv · 2512.02545

On the problem of stability of abstract elementary classes of modules

Gianluca Paolini·Saharon Shelah
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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 $\kappa$-local (also called $\kappa$-tame) for some $\kappa \geq \mathrm{LS}(\mathbf{K}, \preccurlyeq)$, 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 $\kappa$, we have that every abstract elementary class $(\mathbf{K}, \preccurlyeq)$ of $R$-modules with amalgamation satisfying $\kappa > \mathrm{LS}(\mathbf{K}, \preccurlyeq)$ is stable.

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Gianluca Paolini, Saharon Shelah. 2025-12-02. On the problem of stability of abstract elementary classes of modules. https://arxiv.org/abs/2512.02545

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