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Abstract Let R be a Dedekind domain and G a split reductive group, i.e. a Chevalley–Demazure group scheme, of rank ≥2{\geq 2}. We prove that G(R[x1,…,xn])=G(R)E(R[x1,…,xn]) for anyn≥1.G(R[x_{1},\ldots,x_{n}])=G(R)E(R[x_{1},\ldots,x_{n}])\quad\text{for any}\ n% \geq… Click to show full abstract
Abstract Let R be a Dedekind domain and G a split reductive group, i.e. a Chevalley–Demazure group scheme, of rank ≥2{\geq 2}. We prove that G(R[x1,…,xn])=G(R)E(R[x1,…,xn]) for anyn≥1.G(R[x_{1},\ldots,x_{n}])=G(R)E(R[x_{1},\ldots,x_{n}])\quad\text{for any}\ n% \geq 1. In particular, this extends to orthogonal groups the corresponding results of A. Suslin and F. Grunewald, J. Mennicke and L. Vaserstein for G=SLN,Sp2N{G=\mathrm{SL}_{N},\mathrm{Sp}_{2N}}. We also deduce some corollaries of the above result for regular rings R of higher dimension and discrete Hodge algebras over R.
Journal Title: Journal of Group Theory
Year Published: 2019
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