A Northcott type inequality for Buchsbaum-Rim coefficients

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 $b{r}_0(M)-b{r}_1(M)\le \ell (F/M)$, where $b{r}_0(M)$ and $b{r}_1(M$) denote zeroth and first Buchsbaum-Rim coefficients respectively.

16 pages