FPM 1999, vol. 5, no. 3, pp. 943-945

FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

1999, VOLUME 5, NUMBER 3, PAGES 943-945

Symplectic groups over Laurent polynomial rings and patching diagrams

V. I. Kopeiko

Abstract

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In this note we prove the following result. Let $A$
be a P.I.D.\ such that
$\mathop{\mathrm{K}_1\mathrm{Sp}}(A)=0$
. Then the groups
$\mathop{\mathrm{Sp}}_{2r}
(A[X_1^{\pm 1},\ldots,X_n^{\pm 1},Y_1,\ldots,Y_m])$
are generated by elementary symplectic matrices for all integers
$r\geq 2$.

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