Explicit $K_2$ of certain quotient rings over $\mathbb{Z}[G]$ for $G$ an elementary abelian $p$-group

TL;DR

The paper computes the elementary abelian $p$-group structure of $K_2$ for quotients of the group ring $\mathbb{Z}[G]$ with $G$ an elementary abelian $p$-group, using explicit Dennis–Stein symbols and Mayer–Vietoris sequences. It establishes stability and precise ranks for $K_2(\mathbb{Z}[G]/(p^k))$ and $K_2(\mathbb{Z}[G]/p^k\Gamma)$ as $k$ grows, and shows these groups become isomorphic for $k \ge n+1$. The results yield explicit generators and ranks, enabling a definitive description of the relative SK$_1$ groups $SK_1(\mathbb{Z}[G], p^k\mathbb{Z}[G])$ for odd $p$, and connect the K-theory of these quotient rings to classical arithmetic and topological applications. Overall, the work provides concrete, computable invariants for K-theory of group rings of elementary abelian $p$-groups and clarifies the interplay between $K_2$ and relative $SK_1$ in this setting.

Abstract

Let $G$ be an elementary abelian $p$-group. In this paper, we calculate the $K_2$-groups of some quotient rings $\mathbb{Z}[G]/I$ for certain ideals $I \subseteq \mathbb{Z}[G]$ of finite $p$-power index. These results are established through the explicit computation of Dennis-Stein symbols. As an application, we provide a definitive characterization of the relative group $SK_1(\mathbb{Z}[G], p^k\mathbb{Z}[G])$ for any odd prime $p$ and $k \ge 1$.

Theorems & Definitions (23)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Remark 2.5
• Theorem 2.6
• proof
• Lemma 3.1