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

Yakun Zhang

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

Yakun Zhang

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 $r > 1$, let $Γ$ 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 Kähler 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 $r$. Building upon this result, we establish an explicit isomorphism: $$ K_2(\mathbb{Z}[G]/(|G|Γ\cap p\mathbb{Z}[G])) \cong K_2(\mathbb{Z}[G]/|G|Γ) \oplus K_2(\mathbb{F}_p[G], (\widetilde{G})). $$

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

Abstract

This paper investigates the structure of

-groups for certain quotient rings of the integral group ring

. Let

be a finite abelian

-group with

-rank

, let

be the maximal

-order of

, and let

denote the sum of all elements of

in the group ring. By employing the framework of Kähler differentials, we first determine that the relative

-group

is an elementary abelian

-group of rank

. Building upon this result, we establish an explicit isomorphism:

Paper Structure (8 theorems, 24 equations)

This paper contains 8 theorems, 24 equations.

Key Result

Lemma 1.2

alperin1987sk Let $\alpha_0, \dots, \alpha_l$ be elements of a ring such that $1 - \alpha_0 \cdots \alpha_l$ is invertible. Let $\hat{\alpha}_i = \alpha_0 \cdots \alpha_{i-1} \alpha_{i+1} \cdots \alpha_l$. Then

Theorems & Definitions (17)

Remark 1.1
Lemma 1.2
Lemma 1.3
Lemma 2.1
proof
Theorem 2.2
proof
Theorem 2.3
proof
Lemma 3.1
...and 7 more