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

Abstract

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