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

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.