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arXiv · 2401.06330

The abelianization of the elementary group of rank two

Behrooz Mirzaii·Elvis Torres Pérez
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Abstract

For an arbitrary ring $A$, we study the abelianization of the elementary group $\textrm{E}_2(A)$. In particular, we show that for a commutative ring $A$ there exists an exact sequence \[ K_2(2,A)/C(2,A) \to A/M \to \textrm{E}_2(A)^\textrm{ab} \to 1, \] where $C(2,A)$ is the central subgroup of the Steinberg group $\textrm{St}(2,A)$ 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 $x\in A$, $a,b,c \in A^{\times}$.

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Behrooz Mirzaii, Elvis Torres Pérez. 2024-01-12. The abelianization of the elementary group of rank two. https://doi.org/10.1017/s0013091524000877

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