On the $K$-theory of the $p$-adic unit disk

Elden Elmanto; Noah Riggenbach

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arXiv · 2410.07101

On the $K$-theory of the $p$-adic unit disk

Elden Elmanto·Noah Riggenbach

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Abstract

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 \'etale-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.

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BibTeX RIS

Elden Elmanto, Noah Riggenbach. 2024-10-09. On the $K$-theory of the $p$-adic unit disk. https://arxiv.org/abs/2410.07101

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