The classification problem for extensions of torsion-free abelian groups, I
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866915353775308800 |
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| author | Lupini, Martino |
| author_facet | Lupini, Martino |
| contents | Let $C,A$ be countable abelian groups. In this paper we determine the complexity of classifying extensions $C$ by $A$, in the cases when $C$ is torsion-free and $A$ is a $p$-group, a torsion group with bounded primary components, or a free $R$-module for some subring $R\subseteq \mathbb{Q}$. Precisely, for such $C$ and $A$ we describe in terms of $C$ and $A$ the potential complexity class in the sense of Borel complexity theory of the equivalence relation $\mathcal{R}_{\mathbf{Ext}\left( C,A\right) }$ of isomorphism of extensions of $C$ by $A$. This complements a previous result by the same author, settling the case when $C$ is torsion and $A$ is arbitrary. We establish the main result within the framework of Borel-definable homological algebra, recently introduced in collaboration with Bergfalk and Panagiotopoulos. As a consequence of our main results, we will obtain that if $C$ is torsion-free and $A$ is either a free $R$-module or a torsion group with bounded components, then an extension of $C$ by $A$ splits if and only if it splits on all finite-rank subgroups of $C$. This is a purely algebraic statements obtained with methods from Borel-definable homological algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_05431 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | The classification problem for extensions of torsion-free abelian groups, I Lupini, Martino Commutative Algebra Logic 20K35, 54H05 (Primary) 20K40, 20K45 (Secondary) Let $C,A$ be countable abelian groups. In this paper we determine the complexity of classifying extensions $C$ by $A$, in the cases when $C$ is torsion-free and $A$ is a $p$-group, a torsion group with bounded primary components, or a free $R$-module for some subring $R\subseteq \mathbb{Q}$. Precisely, for such $C$ and $A$ we describe in terms of $C$ and $A$ the potential complexity class in the sense of Borel complexity theory of the equivalence relation $\mathcal{R}_{\mathbf{Ext}\left( C,A\right) }$ of isomorphism of extensions of $C$ by $A$. This complements a previous result by the same author, settling the case when $C$ is torsion and $A$ is arbitrary. We establish the main result within the framework of Borel-definable homological algebra, recently introduced in collaboration with Bergfalk and Panagiotopoulos. As a consequence of our main results, we will obtain that if $C$ is torsion-free and $A$ is either a free $R$-module or a torsion group with bounded components, then an extension of $C$ by $A$ splits if and only if it splits on all finite-rank subgroups of $C$. This is a purely algebraic statements obtained with methods from Borel-definable homological algebra. |
| title | The classification problem for extensions of torsion-free abelian groups, I |
| topic | Commutative Algebra Logic 20K35, 54H05 (Primary) 20K40, 20K45 (Secondary) |
| url | https://arxiv.org/abs/2204.05431 |