A Northcott type inequality for Buchsbaum-Rim coefficients

TL;DR

This paper extends Northcott's inequality from Hilbert-Samuel coefficients to Buchsbaum-Rim coefficients for modules over two-dimensional Cohen-Macaulay local rings, establishing a similar bound involving module length.

Contribution

It proves a Northcott type inequality for Buchsbaum-Rim coefficients in the setting of two-dimensional Cohen-Macaulay local rings for modules contained in free modules.

Findings

Established an inequality: br_0(M) - br_1(M) ≤ length(F/M).

Extended Northcott's inequality to Buchsbaum-Rim coefficients.

Applicable to modules in two-dimensional Cohen-Macaulay local rings.

Abstract

In 1960, D.G. Northcott proved that if $e_0(I)$ and $e_1(I)$ denote zeroth and first Hilbert-Samuel coefficients of an $\mathfrak m$-primary ideal $I$ in a Cohen-Macaulay local ring $(R,\mathfrak m)$, then $e_0(I)-e_1(I)\le \ell (R/I)$. In this article, we study an analogue of this inequality for Buchsbaum-Rim coefficients. We prove that if $(R,\mathfrak m)$ is a two dimensional Cohen-Macaulay local ring and $M$ is a finitely generated $R$-module contained in a free module $F$ with finite co-length, then $br_0(M)-br_1(M)\le \ell (F/M)$, where $br_0(M)$ and $br_1(M$) denote zeroth and first Buchsbaum-Rim coefficients respectively.