A Northcott type inequality for Buchsbaum-Rim coefficients

Abstract

In 1960, Northcott \cite {DGN} proved that, if $e_0(I)$ and $e_1(I)$ denote the 0th 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 $\rm{br} _0(M)-\rm{br} _1(M)\le \ell (F/M)$, where $\rm{br} _0(M)$ and $\rm{br} _1(M$) denote 0th and 1st Buchsbaum-Rim coefficients, respectively.