Some remarks on $K_2$ of finite group rings and related algebras
Yakun Zhang
TL;DR
This work analyzes the reduced $K_2$-group of group rings with coefficients in $\mathbb{Z}/p^s\mathbb{Z}$ for finite abelian $p$-groups, and its inverse-limit $p$-adic completion. It derives explicit, computable isomorphisms in two regimes: (i) when the group is cyclic, yielding a formula in terms of $p$-power components and Euler totients, and (ii) for general finite abelian $p$-groups, linking the inverse-limit to cyclic homology $HC_1$. The paper further shows that for the $p$-adic completion $\widehat{\mathbb{Z}}_p[G]$, the inverse limit $\tilde{K}_2^{c}$ is isomorphic to $\bigoplus_{g\in G} G/\langle g\rangle$ and that the natural map into $HC_1$ splits, providing explicit maps and constructions. Overall, the results illuminate the relationship between low-dimensional $K$-theory of group rings and cyclic homology, offering concrete structural descriptions and computable formulas.
Abstract
In this paper, a study is conducted on $K_2$ of the group rings with coefficient in $\mathbb{Z}/p^s\mathbb{Z}$, where the group is a finite abelian $p$-group $G$. The inverse limit of this $K_2$-group is also considered. Furthermore, for the first part of the study, the case where $G$ is a cyclic $p$-group is addressed. Moreover, for the second part, the situation where $G$ is a finite abelian $p$-group is handled. In both cases, an explicit computable isomorphism formula of the corresponding $K_2$-group is provided, and the results are closely related to the cyclic homology groups.