Bounding the $K(p-1)$-local exotic Picard group at $p>3$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911939549986816 |
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| author | Bobkova, Irina Lachmann, Andrea Li, Ang Lima, Alicia Stojanoska, Vesna Zhang, Adela YiYu |
| author_facet | Bobkova, Irina Lachmann, Andrea Li, Ang Lima, Alicia Stojanoska, Vesna Zhang, Adela YiYu |
| contents | In this paper, we bound the descent filtration of the exotic Picard group $κ_n$, for a prime number p>3 and n=p-1. Our method involves a detailed comparison of the Picard spectral sequence, the homotopy fixed point spectral sequence, and an auxiliary $β$-inverted homotopy fixed point spectral sequence whose input is the Farrell-Tate cohomology of the Morava stabilizer group. Along the way, we deduce that the K(n)-local Adams-Novikov spectral sequence for the sphere has a horizontal vanishing line at $3n^2+1$ on the $E_{2n^2+2}$-page. The same analysis also allows us to express the exotic Picard group of $K(n)$-local modules over the homotopy fixed points spectrum $\mathrm{E}_n^{hN}$, where N is the normalizer in $\mathbb{G}_n$ of a finite cyclic subgroup of order p, as a subquotient of a single continuous cohomology group $H^{2n+1}(N,π_{2n}\mathrm{E}_n)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_15572 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounding the $K(p-1)$-local exotic Picard group at $p>3$ Bobkova, Irina Lachmann, Andrea Li, Ang Lima, Alicia Stojanoska, Vesna Zhang, Adela YiYu Algebraic Topology In this paper, we bound the descent filtration of the exotic Picard group $κ_n$, for a prime number p>3 and n=p-1. Our method involves a detailed comparison of the Picard spectral sequence, the homotopy fixed point spectral sequence, and an auxiliary $β$-inverted homotopy fixed point spectral sequence whose input is the Farrell-Tate cohomology of the Morava stabilizer group. Along the way, we deduce that the K(n)-local Adams-Novikov spectral sequence for the sphere has a horizontal vanishing line at $3n^2+1$ on the $E_{2n^2+2}$-page. The same analysis also allows us to express the exotic Picard group of $K(n)$-local modules over the homotopy fixed points spectrum $\mathrm{E}_n^{hN}$, where N is the normalizer in $\mathbb{G}_n$ of a finite cyclic subgroup of order p, as a subquotient of a single continuous cohomology group $H^{2n+1}(N,π_{2n}\mathrm{E}_n)$. |
| title | Bounding the $K(p-1)$-local exotic Picard group at $p>3$ |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2403.15572 |