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| Format: | Preprint |
| Published: |
2025
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| Online Access: | https://arxiv.org/abs/2505.00344 |
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| _version_ | 1866909598089216000 |
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| author | Yang, Tristan |
| author_facet | Yang, Tristan |
| contents | The "higher chromatic" Quillen-Lichtenbaum conjecture, as proposed by Ausoni and Rognes, posits that the finite localization map $K(R) \to L_{n + 1}^f K(R)$ is a $p$-local equivalence in large degrees for suitable ring spectra $R$. We give a simple criterion in terms of syntomic cohomology for an effective version of Quillen-Lichtenbaum, i.e. for identifying the degrees in which the localization map is an isomorphism. Combining our result with recent computations implies that the finite localization map is $(-1)$-truncated in the cases $R = \mathrm{BP} \langle n \rangle$, $R = k(n)$, and $R = \mathrm{ko}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_00344 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Effective Redshift Yang, Tristan Algebraic Topology K-Theory and Homology The "higher chromatic" Quillen-Lichtenbaum conjecture, as proposed by Ausoni and Rognes, posits that the finite localization map $K(R) \to L_{n + 1}^f K(R)$ is a $p$-local equivalence in large degrees for suitable ring spectra $R$. We give a simple criterion in terms of syntomic cohomology for an effective version of Quillen-Lichtenbaum, i.e. for identifying the degrees in which the localization map is an isomorphism. Combining our result with recent computations implies that the finite localization map is $(-1)$-truncated in the cases $R = \mathrm{BP} \langle n \rangle$, $R = k(n)$, and $R = \mathrm{ko}$. |
| title | Effective Redshift |
| topic | Algebraic Topology K-Theory and Homology |
| url | https://arxiv.org/abs/2505.00344 |