A classification of $C_{p^n}$-Tambara fields
Fuente:
arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913720244895744 |
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| author | Wisdom, Noah |
| author_facet | Wisdom, Noah |
| contents | Tambara functors arise in equivariant homotopy theory as the structure adherent to the homotopy groups of a coherently commutative equivariant ring spectrum. We show that if $k$ is a field-like $C_{p^n}$-Tambara functor, then $k$ is the coinduction of a field-like $C_{p^s}$-Tambara functor $\ell$ such that $\ell(C_{p^s}/e)$ is a field. If this field has characteristic other than $p$, we observe that $\ell$ must be a fixed-point Tambara functor, and if the characteristic is $p$, we determine all possible forms of $\ell$ through an analysis of the behavior of the Frobenius endomorphism and the trace of a $C_p$-Galois extension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02966 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A classification of $C_{p^n}$-Tambara fields Wisdom, Noah Algebraic Topology Group Theory Tambara functors arise in equivariant homotopy theory as the structure adherent to the homotopy groups of a coherently commutative equivariant ring spectrum. We show that if $k$ is a field-like $C_{p^n}$-Tambara functor, then $k$ is the coinduction of a field-like $C_{p^s}$-Tambara functor $\ell$ such that $\ell(C_{p^s}/e)$ is a field. If this field has characteristic other than $p$, we observe that $\ell$ must be a fixed-point Tambara functor, and if the characteristic is $p$, we determine all possible forms of $\ell$ through an analysis of the behavior of the Frobenius endomorphism and the trace of a $C_p$-Galois extension. |
| title | A classification of $C_{p^n}$-Tambara fields |
| topic | Algebraic Topology Group Theory |
| url | https://arxiv.org/abs/2409.02966 |