Computations of $K_2$ for certain $\mathbb{Z}\mathrm{/}{p}^s\mathbb{Z}$-algebras and the extension of Oliver's logarithm

This paper describes the $K$-theory structure for three algebra classes. For cyclic $p$-group rings and truncated polynomial rings over $\mathbb{Z}\mathrm{/}{p}^s\mathbb{Z}$, we determine reduced $K_2$-structures via a common algebraic framework. For abelian $p$-group rings over ${\widehat{\mathbb{Z}}}_p$, we extend the isomorphism between reduced continuous $K_2$ and the first cyclic homology group to all finite abelian $p$-groups. A constructive proof using a generalized Artin-Hasse map yields an explicit splitting. This isomorphism is realized by extending Oliver’s $p$-adic logarithm. We also characterize the map from reduced continuous $K_2$ to reduced linearized $K_2$, clarifying the links between $K_2$, cyclic homology, and Kähler differentials.