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NECESSARY AND SUFFICIENT CONDITION FOR CERTAIN SUBSETS OF TRANSVECTIONS TO GENERATE A SUBGROUP OF THE SYMPLECTIC GROUP OVER LOCAL RINGS HIROYUK I ISHIBASHI In memory of Professor Akio Yokoyama 1. Introduction Let V be a free module of rank n over a commutative local ring R with identity 1 and maximal ideal A. Let/ : VxV-> R denote an alternating bilinear form, and let Sp (V) be the symplectic group on / , that is, the set of all linear automorphisms a of V satisfying J[ax, ay) = J{x, y) for all x, y in V. Such a cr is called an isometry on V. Thus Sp(K) is a subgroup of the general linear group GL(F). x L For a submodule U of V we define U = {v e V\f(v, U) = 0} and rad U = U 0 U . For submodules U and W of V, Ul W means U®W with J{U, W) = 0. For any element a in R and any element v in V we define an isometry 7^ „ called a transvection with coefficient a and axis v by the formula: T z = z+J{z,v)-a-v, zeV. av T (S) is the subgroup
Bulletin of the London Mathematical Society – Wiley
Published: Nov 1, 1989
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