The $p$-completed cyclotomic trace in degree $2$

Abstract

We prove that for a quasiregular semiperfectoid ${\mathbb{Z}}_p^{cycl}$-algebra $R$ (in the sense of Bhatt–Morrow–Scholze), the cyclotomic trace map from the $p$-completed $K$-theory spectrum $K\left(R;{\mathbb{Z}}_p\right)$ of $R$ to the topological cyclic homology $TC\left(R;{\mathbb{Z}}_p\right)$ of $R$ identifies on ${\pi }_2$ with a $q$-deformation of the logarithm.