Symmetric powers of complete modules over a two-dimensional regular local ring

Let (R,m) be a two-dimensional regular local ring with infinite residue field. For a finitely generated, torsion-free R-module A, write A(n) for the nth symmetric power of A, mod torsion. We study the modules A(n), n greater than or equal to 1, when A is complete (i.e., integrally closed). In particular, we show that B . A = A(2), for any minimal reduction B subset of or equal to A and that the ring +(n greater than or equal to 1)A(n) is Cohen-Macaulay.