On Hopkins' Picard group
Fuente:
arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | |
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| _version_ | 1866911333982666752 |
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| author | Barthel, Tobias Schlank, Tomer M. Stapleton, Nathaniel Weinstein, Jared |
| author_facet | Barthel, Tobias Schlank, Tomer M. Stapleton, Nathaniel Weinstein, Jared |
| contents | We compute the algebraic Picard group of the category of $K(n)$-local spectra, for all heights $n$ and all primes $p$. In particular, we show that it is always finitely generated over $\mathbb{Z}_p$ and, whenever $n \geq 2$, is of rank $2$, thereby confirming a prediction made by Hopkins in the early 1990s. In fact, with the exception of the anomalous case $n=p=2$, we provide a full set of topological generators for these groups. Our arguments rely on recent advances in $p$-adic geometry to translate the problem to a computation on Drinfeld's symmetric space, which can then be solved using results of Colmez--Dospinescu--Niziol. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20958 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Hopkins' Picard group Barthel, Tobias Schlank, Tomer M. Stapleton, Nathaniel Weinstein, Jared Algebraic Topology Algebraic Geometry Number Theory We compute the algebraic Picard group of the category of $K(n)$-local spectra, for all heights $n$ and all primes $p$. In particular, we show that it is always finitely generated over $\mathbb{Z}_p$ and, whenever $n \geq 2$, is of rank $2$, thereby confirming a prediction made by Hopkins in the early 1990s. In fact, with the exception of the anomalous case $n=p=2$, we provide a full set of topological generators for these groups. Our arguments rely on recent advances in $p$-adic geometry to translate the problem to a computation on Drinfeld's symmetric space, which can then be solved using results of Colmez--Dospinescu--Niziol. |
| title | On Hopkins' Picard group |
| topic | Algebraic Topology Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2407.20958 |