Trivialness of equivariant vector bundles for finite abelian groups

Let G be a finite abelian group, V a complex G-module and A the algebra of all polynomial functions on V. We show that if ill is a free A-module of finite rank, then there exists a G-module W such that M is isomorphic to A circle times W. As a consequence all algebraic G-vector bundles with base V are trivial, and the action of G on the total space of such a bundle is linearizable. 2000 Academic Press.