On the Vasconcelos inequality for the fiber multiplicity of modules

Abstract

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 $\ell (F/M)<\infty$ 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 $f_0(M)\le b{r}_1(M)-b{r}_0(M)+\ell (F/M)+\mu (M)-r$, where $b{r}_i(M)$ denotes the $i^{th}$ Buchsbaum-Rim coefficient of $M$.

18 pages. (To appear in Comm. Algebra)