Lengths and multiplicities of integrally closed modules over a two-dimensional regular local ring

Abstract

Let (R,m) be a two-dimensional regular local ring with infinite residue field. We prove an analogue of the Hoskin–Deligne length formula for a finitely generated, torsion-free, integrally closed R-module M. As a consequence, we get a formula for the Buchsbaum–Rim multiplicity of F/M, where F=M⁎⁎.