On the Vasconcelos inequality for the fiber multiplicity of modules

ABSTRACT

Let (R,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 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,𝔪) is a two-dimensional Cohen–Macaulay local ring, then $f_0\left(M\right)\le b{r}_1\left(M\right)-b{r}_0\left(M\right)+\ell (F∕M)+\mu \left(M\right)-r$, where br_i(M) denotes the i^th Buchsbaum-Rim coefficient of M.

KEYWORDS:

• Buchsbaum-Rim function
• Buchsbaum-Rim polynomial
• fiber cone of modules
• Rees algebra of modules
• Vasconcelos inequality

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D40
• 13A30

Acknowledgments

We sincerely thank the referees for pointing out several errors, some of them typographical and some of them mathematical, which tremendously improved the exposition.