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On symplectic groups over polynomial rings. (English) Zbl 0725.20038
Let A be a locally principal ring, which means that the localization $A\sb{\mu}$ of A at every maximal A-ideal $\mu$ is a principal ideal ring. The principal results of this paper are as follows. Let $R=A[x\sb 1,...,x\sb m]$, where $m\ge 0$. Then (i) $Sp\sb{2n}R=Sp\sb{2n}A\cdot Ep\sb{2n}R$, for all $n\ge 2,$ (ii) $SL\sb nR=SL\sb nA\cdot E\sb nR$, for all $n\ge 3,$ where $Ep\sb{2n}R$ is the subgroup of the symplectic group $Sp\sb{2n}R$ generated by the (symplectic) elementary matrices and $E\sb nR$ is the subgroup of $SL\sb nR$ generated by the elementary matrices (n$\ge 2)$. It follows that $K\sb 1Sp(A)=K\sb 1Sp(R)$ and that $SK\sb 1(A)=SK\sb 1(R)$. These results extend many earlier results of Suslin, Kopejko, Bass and others. The proofs are based on an effective localization and patching technique which reduces the problem to the case where A is a local principal ideal ring. The proofs then make use of symplectic symbols of Mennicke type. Using similar methods the authors also prove some results for the Laurent polynomial ring $R'=A[x,x\sp{-1}]$. When A is a local (principal ideal) ring it is proved that (i) $Sp\sb{2n}R'=Ep\sb{2n}R'$, (ii) $SL\sb{n+1}R'=E\sb{n+1}R'$, for all $n\ge 2$. Finally it is proved that when A is a principal ideal domain $Sp\sb{2n}R'=Sp\sb{2n}A\cdot Ep\sb{2n}R'$, for all $n\ge 2$. It follows that $K\sb 1Sp(A)=K\sb 1Sp(R')$.

##### MSC:
 20H25 Other matrix groups over rings 19B14 Stability for linear groups ($K_1$) 13F10 Principal ideal rings 20G35 Linear algebraic groups over adèles and other rings and schemes 16S34 Group rings (associative rings), Laurent polynomial rings
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##### References:
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