The abelianization of the elementary group of rank two

2024/01/12 by Mirzaii, Behrooz, Pérez, Elvis Torres · 2 citations
#FOS: Mathematics #K-Theory and Homology (math.KT)

paper · doi:10.48550/arxiv.2401.06330

Abstract

For an arbitrary ring A, we study the abelianization of the elementary group \textrmE₂(A). In particular, we show that for a commutative ring A there exists an exact sequence K₂(2,A)/C(2,A) → A/M → \textrmE₂(A)^\textrmab → 1, where C(2,A) is the central subgroup of the Steinberg group \textrmSt(2,A) generated by the Steinberg symbols and M is the additive subgroup of A generated by x(a²-1) and 3(b+1)(c+1), with x∈ A, a,b,c ∈ A^×.

Cited by

• Schur multiplier of SL₂ over finite commutative rings
• Abelianization of SL₂ over Dedekind domains of arithmetic type

Related

• The low dimensional homology groups of the elementary group of degree two
• Abelianization of SL₂ over Dedekind domains of arithmetic type
• The second homology of SL₂ of S-integers
• Explicit K₂ of some quotient rings of ℤ[G], G an elementary abelian p-group
• The abelianization of SL₂(ℤ[(1)/(m)])