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The abelianization of the elementary group of rank two

Part of: Steinberg groups and $K_2$

Published online by Cambridge University Press:  20 January 2025

Behrooz Mirzaii Open the ORCID record for Behrooz Mirzaii [Opens in a new window]  and
Elvis Torres Pérez Open the ORCID record for Elvis Torres Pérez [Opens in a new window]
Behrooz Mirzaii
Affiliation:
Instituto de Ciências Matemáticas e de Computação (ICMC), Universidade de São Paulo, São Carlos, Brazil
Elvis Torres Pérez*
Affiliation:
Factultad de Ciencias, Universidad Nacional de Ingeniería (UNI), Lima, Perú
*
Corresponding author: Elvis Torres Pérez, email: elvis.torres.p@uni.pe
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Abstract

For an arbitrary ring A, we study the abelianization of the elementary group $\mathit{{\rm E}}_2(A)$. In particular, we show that for a commutative ring A there exists an exact sequence

\begin{equation*}{\rm K}_2(2,A)/{\rm C}(2,A) \rightarrow A/M \rightarrow \mathit{{\rm E}}_2(A)^{\rm ab} \rightarrow 1,\end{equation*}

where ${\rm C}(2,A)$ is the central subgroup of the Steinberg group $\mathit{{\rm St}}(2,A)$ generated by the Steinberg symbols and M is the additive subgroup of A generated by $x(a^2-1)$ and $3(b+1)(c+1)$, with $x\in A, a,b,c \in {A^\times}$.

Keywords

algebraic K-theorygroup homologySteinberg symbols

MSC classification

Primary: 19C99: None of the above, but in this section

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 68 , Issue 2 , May 2025 , pp. 487 - 505
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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