<p>In this article, we study the first, second, and third homology groups of the elementary group <InlineEquation ID="IEq1"> <EquationSource Format="TEX">${\textrm{E}}_2(A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>E</mtext> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <i>A</i> is a commutative ring. In particular, we establish a refined Bloch–Wigner type exact sequence over a semilocal ring (subject to mild restrictions on its residue fields) in which either <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$-1 \in ({A^\times })^2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mn>1</mn> <mo>∈</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>×</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$|{A^\times }/({A^\times })^2| \le 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mo stretchy="false">|</mo> </mrow> <msup> <mi>A</mi> <mo>×</mo> </msup> <mo stretchy="false">/</mo> <msup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mo>×</mo> </msup> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mrow> <mo stretchy="false">|</mo> <mo>≤</mo> <mn>4</mn> </mrow> </mrow> </math></EquationSource> </InlineEquation>.</p>

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

期刊论文
发表日期：

2026年3月4日

查看全文

Behrooz Mirzaii,
Elvis Torres Pérez

摘要

In this article, we study the first, second, and third homology groups of the elementary group ${\textrm{E}}_2(A)$ $E_{2} (A)$ , where A is a commutative ring. In particular, we establish a refined Bloch–Wigner type exact sequence over a semilocal ring (subject to mild restrictions on its residue fields) in which either $-1 \in ({A^\times })^2$ $- 1 \in {(A^{\times})}^{2}$ or $|{A^\times }/({A^\times })^2| \le 4$ $| A^{\times} / {(A^{\times})}^{2} | \leq 4$ .