On The Telescopic Picard Group
Fuente:
arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916521551331328 |
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| author | Keidar, Shai |
| author_facet | Keidar, Shai |
| contents | We prove that for any prime $p$ and height $n \ge 1$, the telescopic Picard group $\mathrm{Pic}(\mathrm{Sp}_{Tn})$ contains a subgroup of the form $\mathbb{Z}_p \times \mathbb{Z}/a_p(p^n-1)$, where $a_p = 1$ if $p = 2$ and $a_p = 2$ if $p$ is odd. Using Kummer theory, we obtain an $(\mathbb{F}_{p^n}^\times \rtimes \mathbb{Z}/n)$-Galois extension of $\mathbb{S}_{T(n)}$, obtaining the first example of a lift of a non-Abelian Galois extension of the $K(n)$-local sphere to the telescopic world, at arbitrary positive height and prime. Our proof proceeds by setting up a higher categorical framework for the periodicity theorem, utilizing the symmetries of this framework to construct Picard elements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07716 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On The Telescopic Picard Group Keidar, Shai Algebraic Topology Category Theory We prove that for any prime $p$ and height $n \ge 1$, the telescopic Picard group $\mathrm{Pic}(\mathrm{Sp}_{Tn})$ contains a subgroup of the form $\mathbb{Z}_p \times \mathbb{Z}/a_p(p^n-1)$, where $a_p = 1$ if $p = 2$ and $a_p = 2$ if $p$ is odd. Using Kummer theory, we obtain an $(\mathbb{F}_{p^n}^\times \rtimes \mathbb{Z}/n)$-Galois extension of $\mathbb{S}_{T(n)}$, obtaining the first example of a lift of a non-Abelian Galois extension of the $K(n)$-local sphere to the telescopic world, at arbitrary positive height and prime. Our proof proceeds by setting up a higher categorical framework for the periodicity theorem, utilizing the symmetries of this framework to construct Picard elements. |
| title | On The Telescopic Picard Group |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2412.07716 |