On the $K$-theory of the $p$-adic unit disk
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909343275810816 |
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| author | Elmanto, Elden Riggenbach, Noah |
| author_facet | Elmanto, Elden Riggenbach, Noah |
| contents | In this note, we study the $p$-complete topological cyclic homology of the affine line relative to a ring $A$ which is smooth over a perfectoid ring $R$. Denoting by $NTC(A; \mathbb{Z}_p)$ the spectrum which measures the failure of $\mathbb{A}^1$-invariance on $A$, we observe a kind of Quillen-Lichtenbaum phenomena for $NTC(A; \mathbb{Z}_p)$ -- that it is isomorphic to its own $K(1)$-localization in a specified range of degrees which depends on the relative dimension of $A$. Somewhat surprisingly, this range is better than considerations following from a theorem of Bhatt-Mathew and étale-to-syntomic comparisons. Via the Dundas-Goodwillie-McCarthy theorem, we obtain a description of the algebraic $K$-theory of $p$-completed affine line over such rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_07101 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the $K$-theory of the $p$-adic unit disk Elmanto, Elden Riggenbach, Noah K-Theory and Homology Algebraic Geometry Algebraic Topology In this note, we study the $p$-complete topological cyclic homology of the affine line relative to a ring $A$ which is smooth over a perfectoid ring $R$. Denoting by $NTC(A; \mathbb{Z}_p)$ the spectrum which measures the failure of $\mathbb{A}^1$-invariance on $A$, we observe a kind of Quillen-Lichtenbaum phenomena for $NTC(A; \mathbb{Z}_p)$ -- that it is isomorphic to its own $K(1)$-localization in a specified range of degrees which depends on the relative dimension of $A$. Somewhat surprisingly, this range is better than considerations following from a theorem of Bhatt-Mathew and étale-to-syntomic comparisons. Via the Dundas-Goodwillie-McCarthy theorem, we obtain a description of the algebraic $K$-theory of $p$-completed affine line over such rings. |
| title | On the $K$-theory of the $p$-adic unit disk |
| topic | K-Theory and Homology Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2410.07101 |