Chevalley groups over Laurent polynomial rings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910716581117952 |
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| author | Stavrova, Anastasia |
| author_facet | Stavrova, Anastasia |
| contents | Let $G$ be a simply connected Chevalley--Demazure group scheme without $SL_2$-factors. For any unital commutative ring $R$, we denote by $E(R)$ the standard elementary subgroup of $G(R)$, that is, the subgroup generated by the elementary root unipotent elements. We prove that the map $$ G(R[x_1^{\pm 1},\ldots,x_n^{\pm 1}])/E(R[x_1^{\pm 1},\ldots,x_n^{\pm 1}])\to G\bigl(R((x_1))\ldots((x_n))\bigr)/E\bigl(R((x_1))\ldots((x_n))\bigr) $$ is injective for any $n\ge 1$, if $R$ is either a Dedekind domain or a Noetherian ring that is geometrically regular over a Dedekind domain with perfect residue fields. For $n=1$ this map is also an isomorphism. As a consequence, we show that if $D$ is a PID such that $SL_2(D)=E_2(D)$ (e.g. $D=\mathbb{Z}$), then $G(D[x_1^{\pm 1},\ldots,x_n^{\pm 1}])=E(D[x_1^{\pm 1},\ldots,x_n^{\pm 1}])$. This extends earlier results for special linear and symplectic groups due to A. A. Suslin and V. I. Kopeiko. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_17308 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Chevalley groups over Laurent polynomial rings Stavrova, Anastasia Group Theory K-Theory and Homology Let $G$ be a simply connected Chevalley--Demazure group scheme without $SL_2$-factors. For any unital commutative ring $R$, we denote by $E(R)$ the standard elementary subgroup of $G(R)$, that is, the subgroup generated by the elementary root unipotent elements. We prove that the map $$ G(R[x_1^{\pm 1},\ldots,x_n^{\pm 1}])/E(R[x_1^{\pm 1},\ldots,x_n^{\pm 1}])\to G\bigl(R((x_1))\ldots((x_n))\bigr)/E\bigl(R((x_1))\ldots((x_n))\bigr) $$ is injective for any $n\ge 1$, if $R$ is either a Dedekind domain or a Noetherian ring that is geometrically regular over a Dedekind domain with perfect residue fields. For $n=1$ this map is also an isomorphism. As a consequence, we show that if $D$ is a PID such that $SL_2(D)=E_2(D)$ (e.g. $D=\mathbb{Z}$), then $G(D[x_1^{\pm 1},\ldots,x_n^{\pm 1}])=E(D[x_1^{\pm 1},\ldots,x_n^{\pm 1}])$. This extends earlier results for special linear and symplectic groups due to A. A. Suslin and V. I. Kopeiko. |
| title | Chevalley groups over Laurent polynomial rings |
| topic | Group Theory K-Theory and Homology |
| url | https://arxiv.org/abs/2411.17308 |