Buchsbaum-Rim multiplicities as Hilbert-Samuel multiplicities

We study the Buchsbaum-Rim multiplicity br (M) of a finitely generated module M over a regular local ring R of dimension 2 with maximal ideal m. The module M under consideration is of finite colength in a free R-module F. Write F / M ≅ I / J, where J ⊂ I are m-primary ideals of R. We first investigate the colength ℓ (R / a) of any m-primary ideal a and its Hilbert-Samuel multiplicity e (a) using linkage theory. As an application, we establish several multiplicity formulas that express the Buchsbaum-Rim multiplicity of the module M in terms of the Hilbert-Samuel multiplicities of ideals related to I, J and a minimal reduction of M. The motivation comes from work by E. Jones, who applied graphical computations of the Hilbert-Samuel multiplicity to the Buchsbaum-Rim multiplicity [E. Jones, Computations of Buchsbaum-Rim multiplicities, J. Pure Appl. Algebra 162 (2001) 37-52].