Schur multiplier of $\mathrm{SL}_2$ over finite commutative rings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914075671265280 |
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| author | Mirzaii, Behrooz Vega, Abraham Rojas |
| author_facet | Mirzaii, Behrooz Vega, Abraham Rojas |
| contents | In this article, we investigate the Schur multiplier of the special linear group $\mathrm{SL}_2(A)$ over finite commutative local rings $A$. We prove that the Schur multiplier of these groups is isomorphic to the $K$-group $K_2(A)$ whenever the residue field $A/\mathfrak{m}_A$ has odd characteristic and satisfies $|A/\mathfrak{m}_A| \neq 3,5,9$. As an application, we show that if $A$ is either the Galois ring $\mathrm{GR}(p^l,m)$ or the quasi-Galois ring $A(p^m,n)$ with residue field of odd characteristic and $|A/\mathfrak{m}_A| \neq 3,5,9$, then the Schur multiplier of $\mathrm{SL}_2(A)$ is trivial. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_03946 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Schur multiplier of $\mathrm{SL}_2$ over finite commutative rings Mirzaii, Behrooz Vega, Abraham Rojas K-Theory and Homology Group Theory 19C09, 19B14, 20J06 In this article, we investigate the Schur multiplier of the special linear group $\mathrm{SL}_2(A)$ over finite commutative local rings $A$. We prove that the Schur multiplier of these groups is isomorphic to the $K$-group $K_2(A)$ whenever the residue field $A/\mathfrak{m}_A$ has odd characteristic and satisfies $|A/\mathfrak{m}_A| \neq 3,5,9$. As an application, we show that if $A$ is either the Galois ring $\mathrm{GR}(p^l,m)$ or the quasi-Galois ring $A(p^m,n)$ with residue field of odd characteristic and $|A/\mathfrak{m}_A| \neq 3,5,9$, then the Schur multiplier of $\mathrm{SL}_2(A)$ is trivial. |
| title | Schur multiplier of $\mathrm{SL}_2$ over finite commutative rings |
| topic | K-Theory and Homology Group Theory 19C09, 19B14, 20J06 |
| url | https://arxiv.org/abs/2510.03946 |