Koszul Resolutions over Free Incomplete Tambara Functors for Cyclic $p$-Groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914888163524608 |
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| author | Mehrle, David Quigley, J. D. Stahlhauer, Michael |
| author_facet | Mehrle, David Quigley, J. D. Stahlhauer, Michael |
| contents | In equivariant algebra, Mackey functors replace abelian groups and incomplete Tambara functors replace commutative rings. In this context, we prove that equivariant Hochschild homology can sometimes be computed using Mackey functor-valued Tor. To compute these Tor Mackey functors for odd primes $p$, we define cyclic-$p$-group-equivariant analogues of the Koszul resolution which resolve the Burnside Mackey functor (the analogue of the integers) as a module over free incomplete Tambara functors (the analogue of polynomial rings). We apply these Koszul resolutions to compute Mackey functor-valued Hochschild homology of free incomplete Tambara functors for cyclic groups of odd prime order and for the cyclic group of order 9. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_18382 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Koszul Resolutions over Free Incomplete Tambara Functors for Cyclic $p$-Groups Mehrle, David Quigley, J. D. Stahlhauer, Michael Algebraic Topology Group Theory In equivariant algebra, Mackey functors replace abelian groups and incomplete Tambara functors replace commutative rings. In this context, we prove that equivariant Hochschild homology can sometimes be computed using Mackey functor-valued Tor. To compute these Tor Mackey functors for odd primes $p$, we define cyclic-$p$-group-equivariant analogues of the Koszul resolution which resolve the Burnside Mackey functor (the analogue of the integers) as a module over free incomplete Tambara functors (the analogue of polynomial rings). We apply these Koszul resolutions to compute Mackey functor-valued Hochschild homology of free incomplete Tambara functors for cyclic groups of odd prime order and for the cyclic group of order 9. |
| title | Koszul Resolutions over Free Incomplete Tambara Functors for Cyclic $p$-Groups |
| topic | Algebraic Topology Group Theory |
| url | https://arxiv.org/abs/2407.18382 |