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

TL;DR

This paper investigates the Buchsbaum-Rim function of a parameter module, establishing an upper bound, characterizing Cohen-Macaulay rings, and analyzing the sign of the first Buchsbaum-Rim coefficient.

Contribution

It provides a bound for the Buchsbaum-Rim function and links the equality case to Cohen-Macaulay rings, also analyzing the sign of the first coefficient.

Findings

Buchsbaum-Rim function is bounded above by a specific binomial coefficient expression.

Equality in the bound characterizes Cohen-Macaulay rings.

The first Buchsbaum-Rim coefficient is always non-positive.

Abstract

In this article, we prove that the Buchsbaum-Rim function $\ell_A(\S_{\nu+1}(F)/N^{\nu+1})$ of a parameter module $N$ in $F$ is bounded above by $e(F/N) \binom{\nu+d+r-1}{d+r-1}$ for every integer $\nu \geq 0$. Moreover, it turns out that the base ring $A$ is Cohen-Macaulay once the equality holds for some integer $\nu$. As a direct consequence, we observe that the first Buchsbaum-Rim coefficient $e_1(F/N)$ of a parameter module $N$ is always non-positive.