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

Let (R,m)(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/MF/M, where F=M⁎⁎F=M⁎⁎.