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A Northcott type inequality for Buchsbaum-Rim coefficients. (English) Zbl 1358.13007

Summary: In [J. Lond. Math. Soc. 35, 209–214 (1960; Zbl 0118.04502)] D. G. Northcott 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 \(e_0(I)-e_1(I)\leq \ell (R/I)\). 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 \(\mathrm{br}_0(M)-\mathrm{br}_1(M)\leq \ell (F/M)\), where \(\mathrm{br}_0(M)\) and \(\mathrm{br}_1(M\)) denote 0th and 1st Buchsbaum-Rim coefficients, respectively.

MSC:

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

Citations:

Zbl 0118.04502
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