On the problem of stability of abstract elementary classes of modules

TL;DR

The paper investigates stability in abstract elementary classes of modules, providing counterexamples and establishing conditions under which stability or almost stability holds, including results assuming large cardinal axioms.

Contribution

It constructs unstable classes of torsion-free abelian groups and introduces the notion of almost stability, proving stability results under tameness and large cardinal assumptions.

Findings

Counterexamples of unstable classes of torsion-free abelian groups

Introduction of the concept of almost stability

Stability results assuming tameness and large cardinals

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.