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

Abstract

Let ( R , m ) (R,m) be a two-dimensional regular local ring with infinite residue field. For a finitely generated, torsion-free R R -module A A , write A n A_{n} for the n n th symmetric power of A A , mod torsion. We study the modules A n A_{n} , n ≥ 1 n \geq 1 , when A A is complete (i.e., integrally closed). In particular, we show that B ⋅ A = A 2 B\cdot A = A_{2} , for any minimal reduction B ⊆ A B \subseteq A and that the ring ⊕ n ≥ 1 A n \oplus _{n \geq 1} A_{n} is Cohen-Macaulay.