Homological algebra of pro-Lie Polish abelian groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913645301071872 |
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| author | Casarosa, Matteo Codenotti, Alessandro Lupini, Martino |
| author_facet | Casarosa, Matteo Codenotti, Alessandro Lupini, Martino |
| contents | In this paper, we initiate the study of pro-Lie Polish abelian groups from the perspective of homological algebra. We extend to this context the type-decomposition of locally compact Polish abelian groups of Hoffmann and Spitzweck, and prove that the category $\mathbf{proLiePAb}$ of pro-Lie Polish abelian groups is a thick subcategory of the category of Polish abelian groups. We completely characterize injective and projective objects in $\mathbf{proLiePAb}$. We conclude that $\mathbf{proLiePAb}$ has enough projectives but not enough injectives and homological dimension $1$. We also completely characterize injective and projective objects in the category of non-Archimedean Polish abelian groups, concluding that it has enough injectives and projectives and homological dimension $1$. Injective objects are also characterized for the categories of topological torsion Polish abelian groups and for Polish abelian topological $p$-groups, showing that these categories have enough injectives and homological dimension $1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13444 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Homological algebra of pro-Lie Polish abelian groups Casarosa, Matteo Codenotti, Alessandro Lupini, Martino Commutative Algebra Category Theory Logic 54H05, 20K45, 18F60, 26E30, 18G10, 46M15 In this paper, we initiate the study of pro-Lie Polish abelian groups from the perspective of homological algebra. We extend to this context the type-decomposition of locally compact Polish abelian groups of Hoffmann and Spitzweck, and prove that the category $\mathbf{proLiePAb}$ of pro-Lie Polish abelian groups is a thick subcategory of the category of Polish abelian groups. We completely characterize injective and projective objects in $\mathbf{proLiePAb}$. We conclude that $\mathbf{proLiePAb}$ has enough projectives but not enough injectives and homological dimension $1$. We also completely characterize injective and projective objects in the category of non-Archimedean Polish abelian groups, concluding that it has enough injectives and projectives and homological dimension $1$. Injective objects are also characterized for the categories of topological torsion Polish abelian groups and for Polish abelian topological $p$-groups, showing that these categories have enough injectives and homological dimension $1$. |
| title | Homological algebra of pro-Lie Polish abelian groups |
| topic | Commutative Algebra Category Theory Logic 54H05, 20K45, 18F60, 26E30, 18G10, 46M15 |
| url | https://arxiv.org/abs/2405.13444 |