On the Vasconcelos inequality for the fiber multiplicity of modules

Let (R,&) be a Noetherian local ring of dimension d>0 with infinite residue field. Let M be a finitely generated proper R-submodule of a free R-module F with ℓ(F∕M)<∞ and having rank r. In this article, we study the fiber multiplicity f0(M) of the module M. We prove that if (R,&) is a two-dimensional Cohen–Macaulay local ring, then f0(M)≤br1(M)-br0(M)+l(F/M)+ m (M)-r, where bri(M) denotes the ith Buchsbaum-Rim coefficient of M.