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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.23115 |
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| _version_ | 1866912633897091072 |
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| author | Meyer, Samuel Wagner, Ferdinand |
| author_facet | Meyer, Samuel Wagner, Ferdinand |
| contents | As a consequence of Efimov's proof of rigidity of the $\infty$-category of localising motives, Efimov and Scholze have constructed refinements of localising invariants such as $\operatorname{THH}$ and $\operatorname{TC}^-$. These refinements often contain vastly more information than the original invariant. In this article we explain a general recipe how to compute the refinements in certain situations. We then apply this recipe to compute the homotopy groups of $\operatorname{TC}^{-,\mathrm{ref}}(\mathrm{ku}\otimes\mathbb Q/\mathrm{ku})$ and $\operatorname{TC}^{-,\mathrm{ref}}(\mathrm{KU}\otimes\mathbb Q/\mathrm{KU})$. The result has a rather surprising geometric description and contains non-trivial information modulo any prime, in contrast to the unrefined $\operatorname{TC}^-$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_23115 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $q$-Hodge complexes and refined $\operatorname{TC}^-$ Meyer, Samuel Wagner, Ferdinand Algebraic Topology Algebraic Geometry 14F30, 14F40, 14G25 (Primary), 19D55, 55P42, 55P43 (Secondary) As a consequence of Efimov's proof of rigidity of the $\infty$-category of localising motives, Efimov and Scholze have constructed refinements of localising invariants such as $\operatorname{THH}$ and $\operatorname{TC}^-$. These refinements often contain vastly more information than the original invariant. In this article we explain a general recipe how to compute the refinements in certain situations. We then apply this recipe to compute the homotopy groups of $\operatorname{TC}^{-,\mathrm{ref}}(\mathrm{ku}\otimes\mathbb Q/\mathrm{ku})$ and $\operatorname{TC}^{-,\mathrm{ref}}(\mathrm{KU}\otimes\mathbb Q/\mathrm{KU})$. The result has a rather surprising geometric description and contains non-trivial information modulo any prime, in contrast to the unrefined $\operatorname{TC}^-$. |
| title | $q$-Hodge complexes and refined $\operatorname{TC}^-$ |
| topic | Algebraic Topology Algebraic Geometry 14F30, 14F40, 14G25 (Primary), 19D55, 55P42, 55P43 (Secondary) |
| url | https://arxiv.org/abs/2410.23115 |