On the problem of stability of abstract elementary classes of modules
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866914503086571520 |
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| author | Paolini, Gianluca Shelah, Saharon |
| author_facet | Paolini, Gianluca Shelah, Saharon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_02545 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the problem of stability of abstract elementary classes of modules Paolini, Gianluca Shelah, Saharon Logic 03C48, 03C45, 13L05, 16D10 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. |
| title | On the problem of stability of abstract elementary classes of modules |
| topic | Logic 03C48, 03C45, 13L05, 16D10 |
| url | https://arxiv.org/abs/2512.02545 |