On the Vasconcelos inequality for the fiber multiplicity of modules

TL;DR

This paper establishes an inequality relating the fiber multiplicity of modules over two-dimensional Cohen-Macaulay local rings, involving Buchsbaum-Rim coefficients and module invariants, extending the Vasconcelos inequality.

Contribution

It proves a new inequality for fiber multiplicity of modules over Cohen-Macaulay rings, generalizing Vasconcelos inequality using Buchsbaum-Rim coefficients.

Findings

Established an upper bound for fiber multiplicity in terms of Buchsbaum-Rim coefficients.

Extended Vasconcelos inequality to modules over two-dimensional Cohen-Macaulay rings.

Provided a new relation connecting module invariants and multiplicity measures.

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 $f_0(M)\le br_1(M)-br_0(M)+\ell (F/M)+\mu(M)-r$, where $br_i(M)$ denotes the $i^{th}$ Buchsbaum-Rim coefficient of $M$.