A motivic filtration on the topological cyclic homology of commutative ring spectra
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| Format: | Preprint |
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2022
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| _version_ | 1866911218541789184 |
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| author | Hahn, Jeremy Raksit, Arpon Wilson, Dylan |
| author_facet | Hahn, Jeremy Raksit, Arpon Wilson, Dylan |
| contents | For a prime number $p$ and a $p$-quasisyntomic commutative ring $R$, Bhatt--Morrow--Scholze defined motivic filtrations on the $p$-completions of $\mathrm{THH}(R), \mathrm{TC}^{-}(R), \mathrm{TP}(R),$ and $\mathrm{TC}(R)$, with the associated graded objects for $\mathrm{TP}(R)$ and $\mathrm{TC}(R)$ recovering the prismatic and syntomic cohomology of $R$, respectively. We give an alternate construction of these filtrations that applies also when $R$ is a well-behaved commutative ring spectrum; for example, we can take $R$ to be $\mathbb{S}$, $\mathrm{MU}$, $\mathrm{ku}$, $\mathrm{ko}$, or $\mathrm{tmf}$. We compute the mod $(p,v_1)$ syntomic cohomology of the Adams summand $\ell$ and observe that, when $p \ge 3$, the motivic spectral sequence for $V(1)_*\mathrm{TC}(\ell)$ collapses at the $\mathrm{E}_2$-page. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2206_11208 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A motivic filtration on the topological cyclic homology of commutative ring spectra Hahn, Jeremy Raksit, Arpon Wilson, Dylan K-Theory and Homology Algebraic Geometry Algebraic Topology For a prime number $p$ and a $p$-quasisyntomic commutative ring $R$, Bhatt--Morrow--Scholze defined motivic filtrations on the $p$-completions of $\mathrm{THH}(R), \mathrm{TC}^{-}(R), \mathrm{TP}(R),$ and $\mathrm{TC}(R)$, with the associated graded objects for $\mathrm{TP}(R)$ and $\mathrm{TC}(R)$ recovering the prismatic and syntomic cohomology of $R$, respectively. We give an alternate construction of these filtrations that applies also when $R$ is a well-behaved commutative ring spectrum; for example, we can take $R$ to be $\mathbb{S}$, $\mathrm{MU}$, $\mathrm{ku}$, $\mathrm{ko}$, or $\mathrm{tmf}$. We compute the mod $(p,v_1)$ syntomic cohomology of the Adams summand $\ell$ and observe that, when $p \ge 3$, the motivic spectral sequence for $V(1)_*\mathrm{TC}(\ell)$ collapses at the $\mathrm{E}_2$-page. |
| title | A motivic filtration on the topological cyclic homology of commutative ring spectra |
| topic | K-Theory and Homology Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2206.11208 |