One Decomposition of $K_2$-Group for Certain Quotients over $\mathbb{Z}[G]$ with $G$ a Finite Abelian $p$-Group

TL;DR

This paper provides a detailed decomposition of the $K_2$-group for certain quotients of the integral group ring of finite abelian $p$-groups, revealing explicit structures and isomorphisms using Kähler differentials.

Contribution

It introduces a novel decomposition of $K_2$-groups for specific quotients of $bZ[G]$, connecting them to elementary abelian $p$-groups and establishing explicit isomorphisms.

Findings

$K_2(bF_p[G], ( ilde{G}))$ is an elementary abelian $p$-group of rank $r$ for $|G|>2$

An explicit isomorphism for $K_2$ of certain quotient rings of $bZ[G]$ is established

The structure of $K_2$-groups is clarified using Kähler differentials and group ring properties

Abstract

This paper investigates the structure of $K_2$-groups for certain quotient rings of the integral group ring $\mathbb{Z}[G]$. Let $G$ be a finite abelian $p$-group with $p$-rank $r$, let $\Gamma$ be the maximal $\mathbb{Z}$-order of $\mathbb{Q}[G]$, and let $\widetilde{G}$ denote the sum of all elements of $G$ in the group ring. By employing the framework of K\"{a}hler differentials, we first determine that the relative $K$-group $K_2(\mathbb{F}_p[G], (\widetilde{G}))$ is an elementary abelian $p$-group of rank $r$ when $|G|>2$. Building upon this result, we establish an explicit isomorphism for $r > 1$: $$ K_2(\mathbb{Z}[G]/(|G|\Gamma \cap p\mathbb{Z}[G])) \cong K_2(\mathbb{Z}[G]/|G|\Gamma) \oplus K_2(\mathbb{F}_p[G], (\widetilde{G})). $$