The abelianization of the elementary group of rank two
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916807812579328 |
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| author | Mirzaii, Behrooz Pérez, Elvis Torres |
| author_facet | Mirzaii, Behrooz Pérez, Elvis Torres |
| contents | 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 $x\in A$, $a,b,c \in A^{\times}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06330 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The abelianization of the elementary group of rank two Mirzaii, Behrooz Pérez, Elvis Torres K-Theory and Homology 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 $x\in A$, $a,b,c \in A^{\times}$. |
| title | The abelianization of the elementary group of rank two |
| topic | K-Theory and Homology |
| url | https://arxiv.org/abs/2401.06330 |