Applications of Borel-definable homological algebra to locally compact groups
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912615864729600 |
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| author | Lupini, Martino |
| author_facet | Lupini, Martino |
| contents | We show that the $\mathrm{Hom}$ functor from the category $\mathbf{LCPAb}$ of locally compact Polish abelian groups to the category $\mathbf{PAb}$ of Polish abelian groups has a total right derived functor, improving on Hoffmann and Spitzweck's construction of its cohomological right derived functor.
We also apply the description of the left heart of subcategories of $\mathbf{PAb}$ in terms of groups with a Polish cover and Borel-definable group homomorphisms to completely characterize the injective and projective objects in the left heart of $\mathbf{LCPAb}$, as well as in the left heart of its full subcategories spanned by: compactly generated groups, Lie groups, totally disconnected groups, topological torsion groups, topological $p$-groups, locally compact groups of finite ranks, topological torsion groups of finite ranks, topological $p$-groups of finite ranks, and type $\mathbb{A}$ groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_25474 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Applications of Borel-definable homological algebra to locally compact groups Lupini, Martino Group Theory Category Theory General Topology Logic 54H05, 20K45, 18F60 (Primary) 26E30, 18G10, 46M15 (Secondary) We show that the $\mathrm{Hom}$ functor from the category $\mathbf{LCPAb}$ of locally compact Polish abelian groups to the category $\mathbf{PAb}$ of Polish abelian groups has a total right derived functor, improving on Hoffmann and Spitzweck's construction of its cohomological right derived functor. We also apply the description of the left heart of subcategories of $\mathbf{PAb}$ in terms of groups with a Polish cover and Borel-definable group homomorphisms to completely characterize the injective and projective objects in the left heart of $\mathbf{LCPAb}$, as well as in the left heart of its full subcategories spanned by: compactly generated groups, Lie groups, totally disconnected groups, topological torsion groups, topological $p$-groups, locally compact groups of finite ranks, topological torsion groups of finite ranks, topological $p$-groups of finite ranks, and type $\mathbb{A}$ groups. |
| title | Applications of Borel-definable homological algebra to locally compact groups |
| topic | Group Theory Category Theory General Topology Logic 54H05, 20K45, 18F60 (Primary) 26E30, 18G10, 46M15 (Secondary) |
| url | https://arxiv.org/abs/2509.25474 |