The low dimensional homology groups of the elementary group of degree two

TL;DR

This paper investigates the low-dimensional homology groups of the elementary group of degree two over a commutative ring, providing new exact sequences under specific algebraic conditions.

Contribution

It introduces a refined Bloch-Wigner type exact sequence for the homology groups of (A) over semilocal rings with particular residue field restrictions.

Findings

Established the structure of the first, second, and third homology groups of (A).

Derived a refined exact sequence generalizing previous results.

Applied the sequence to rings with specific units and residue field conditions.

Abstract

In this article we study the first, the second and the third homology groups of the elementary group $\textrm{E}_2(A)$, where $A$ is a commutative ring. In particular, we prove a refined Bloch-Wigner type exact sequence over a semilocal ring (with some mild restriction on its residue fields) such that $-1\in (A^{\times})^2$ or $|A^{\times}/(A^{\times})^2|\leq 4$.