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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Online-Zugang: | https://arxiv.org/abs/2405.03634 |
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| _version_ | 1866914436079419392 |
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| author | Gheorghiu, Max |
| author_facet | Gheorghiu, Max |
| contents | Viewing group cohomology as a cohomological functor, G. Mislin has generalised Tate cohomology from finite groups to all discrete groups by defining a completion for cohomological functors in 1994. In a previous paper, we have constructed for a cohomological functor $T^{\bullet}: \mathcal{C} \rightarrow \mathcal{D}$ its Mislin completion $\widehat{T}^{\bullet}: \mathcal{C} \rightarrow \mathcal{D}$ under mild assumptions on the abelian categories $\mathcal{C}$ and $\mathcal{D}$, which generalises Tate cohomology to all $T1$ topological groups. In this paper, we investigate the properties of Mislin completions. As their main feature, Mislin completions of Ext-functors detect finite projective dimension of objects in the domain category. We establish a version of dimension shifting, an Eckmann--Shapiro result as well as cohomology products such as external products, cup products and Yoneda products. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_03634 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a Completion of Cohomological Functors Generalising Tate Cohomology II Gheorghiu, Max Group Theory 20J06, 18G15 Viewing group cohomology as a cohomological functor, G. Mislin has generalised Tate cohomology from finite groups to all discrete groups by defining a completion for cohomological functors in 1994. In a previous paper, we have constructed for a cohomological functor $T^{\bullet}: \mathcal{C} \rightarrow \mathcal{D}$ its Mislin completion $\widehat{T}^{\bullet}: \mathcal{C} \rightarrow \mathcal{D}$ under mild assumptions on the abelian categories $\mathcal{C}$ and $\mathcal{D}$, which generalises Tate cohomology to all $T1$ topological groups. In this paper, we investigate the properties of Mislin completions. As their main feature, Mislin completions of Ext-functors detect finite projective dimension of objects in the domain category. We establish a version of dimension shifting, an Eckmann--Shapiro result as well as cohomology products such as external products, cup products and Yoneda products. |
| title | On a Completion of Cohomological Functors Generalising Tate Cohomology II |
| topic | Group Theory 20J06, 18G15 |
| url | https://arxiv.org/abs/2405.03634 |