A computation of $THH_*(ku)$ using a gathered spectral sequence
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910227274661888 |
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| author | Chaminadour, Maxime |
| author_facet | Chaminadour, Maxime |
| contents | In this article, we extend the computation of topological Hochschild homology (THH) of the Adams summand $\ell$ of $p$-local connective complex topological K-theory ($ku$) to $ku$ itself. We leverage the relation $u^{p-1} = v_1$, where $u$ is a generator of $ku_*$ and $v_1$ is a generator of $\ell_*$, and we consider the cofiber of the multiplication by $v_1$ in $ku$, denoted $ku/v_1$. We use the morphism between the Bockstein spectral sequences of the multiplication by $v_1$ computing $THH_*(\ell)$ and $THH_*(ku)$; we develop a general technique using what we term a gathered spectral sequence that allows us to explore the relationship between the Bockstein spectral sequences for the multiplications by $v_1$ and $u$, from which we derive a complete computation of $THH_*(ku)$. Our method is not only applicable to this specific problem but may also prove useful in other computations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_02346 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A computation of $THH_*(ku)$ using a gathered spectral sequence Chaminadour, Maxime Algebraic Topology K-Theory and Homology 55T05 (primary) 55N15 55P42 (secondary) In this article, we extend the computation of topological Hochschild homology (THH) of the Adams summand $\ell$ of $p$-local connective complex topological K-theory ($ku$) to $ku$ itself. We leverage the relation $u^{p-1} = v_1$, where $u$ is a generator of $ku_*$ and $v_1$ is a generator of $\ell_*$, and we consider the cofiber of the multiplication by $v_1$ in $ku$, denoted $ku/v_1$. We use the morphism between the Bockstein spectral sequences of the multiplication by $v_1$ computing $THH_*(\ell)$ and $THH_*(ku)$; we develop a general technique using what we term a gathered spectral sequence that allows us to explore the relationship between the Bockstein spectral sequences for the multiplications by $v_1$ and $u$, from which we derive a complete computation of $THH_*(ku)$. Our method is not only applicable to this specific problem but may also prove useful in other computations. |
| title | A computation of $THH_*(ku)$ using a gathered spectral sequence |
| topic | Algebraic Topology K-Theory and Homology 55T05 (primary) 55N15 55P42 (secondary) |
| url | https://arxiv.org/abs/2511.02346 |