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].