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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
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 
$x\in A, a,b,c \in {A^\times}$.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2025. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.
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