A Note on the Buchsbaum-Rim multiplicity of a parameter module

TL;DR

This paper investigates the Buchsbaum-Rim multiplicity of parameter modules in free modules over a ring, establishing bounds and characterizing Cohen-Macaulay rings through equality conditions.

Contribution

It proves an upper bound for the Buchsbaum-Rim multiplicity and characterizes Cohen-Macaulay rings via equality with colength.

Findings

Buchsbaum-Rim multiplicity is bounded above by colength.

Equality of multiplicity and colength characterizes Cohen-Macaulay rings.

Provides new insights into the structure of parameter modules.

Abstract

In this article we prove that the Buchsbaum-Rim multiplicity $e(F/N)$ of a parameter module $N$ in a free module $F=A^r$ is bounded above by the colength $\ell_A(F/N)$. Moreover, we prove that once the equality $\ell_A(F/N)=e(F/N)$ holds true for some parameter module $N$ in $F$, then the base ring $A$ is Cohen-Macaulay.