Failure of singular compactness for Hom
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915782607241216 |
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| author | Asgharzadeh, Mohsen Golshani, Mohammad Shelah, Saharon |
| author_facet | Asgharzadeh, Mohsen Golshani, Mohammad Shelah, Saharon |
| contents | Assuming Gödel's axiom of constructibility $V=L$, we construct a $χ$-free abelian group $G$ of singular cardinality for some suitable cardinal $χ$ which is regular and uncountable, equipped with the property that for every nontrivial subgroup $G' \subseteq G$ of smaller cardinality, $Hom(G',\mathbb{Z}) \neq 0$, while $Hom(G,\mathbb{Z}) = 0$. This provides a consistent counterexample to the singular compactness of nontrivial duality with respect to the functor $Hom(-,\mathbb{Z})$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_03633 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Failure of singular compactness for Hom Asgharzadeh, Mohsen Golshani, Mohammad Shelah, Saharon Group Theory Commutative Algebra Logic Assuming Gödel's axiom of constructibility $V=L$, we construct a $χ$-free abelian group $G$ of singular cardinality for some suitable cardinal $χ$ which is regular and uncountable, equipped with the property that for every nontrivial subgroup $G' \subseteq G$ of smaller cardinality, $Hom(G',\mathbb{Z}) \neq 0$, while $Hom(G,\mathbb{Z}) = 0$. This provides a consistent counterexample to the singular compactness of nontrivial duality with respect to the functor $Hom(-,\mathbb{Z})$. |
| title | Failure of singular compactness for Hom |
| topic | Group Theory Commutative Algebra Logic |
| url | https://arxiv.org/abs/2506.03633 |