Journal of Commutative Algebra

A Northcott type inequality for Buchsbaum-Rim coefficients

R. Balakrishnan and A.V. Jayanthan

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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.

Article information

J. Commut. Algebra, Volume 8, Number 4 (2016), 493-512.

First available in Project Euclid: 27 October 2016

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Zentralblatt MATH identifier

Primary: 13A30: Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series

Buchsbaum-Rim function Buchsbaum-Rim polynomial Northcott inequality Rees algebra of modules


Balakrishnan, R.; Jayanthan, A.V. A Northcott type inequality for Buchsbaum-Rim coefficients. J. Commut. Algebra 8 (2016), no. 4, 493--512. doi:10.1216/JCA-2016-8-4-493.

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