On the Vasconcelos inequality for the fiber multiplicity of modules

Let $(R,\mathfrak{m})$ 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 $\mathrm{\ell }(F\mathrm{/}M)<\mathrm{\infty }$ and having rank $r$. In this article, we study the fiber multiplicity $f_0(M)$ of the module $M$. We prove that if $(R,\mathfrak{m})$ is a two dimensional Cohen-Macaulay local ring, then $f_0(M)\le b{r}_1(M)-b{r}_0(M)+\mathrm{\ell }(F\mathrm{/}M)+\mu (M)-r$, where $b{r}_i(M)$ denotes the $i^{th}$ Buchsbaum-Rim coefficient of $M$.