Generalized Hilbert coefficients and Northcott's inequality

JOURNAL OF ALGEBRA

卷:461

Article

Xie, Yu¹ 

[1] Penn State Altoona, Dept Math & Stat, Altoona, PA 16601 USA

关键词: Generalized Hilbert coefficients;    Associated graded rings;    Depth;    Cohen-Macaulay;    Northcott's inequality;   

DOI  :  10.1016/j.jalgebra.2016.05.009

来源: Elsevier

【 摘 要 】

Let R be a Cohen-Macaulay local ring of dimension d with infinite residue field. Let I be an R-ideal that has analytic spread l(I) = d, satisfies the G(d) condition and the weak Artin-Nagata property AN(d-2)(-). We provide a formula relating the length lambda(In+1/JI(n)) to the difference P-I(n)-H-I(n), where J is a general minimal reduction of I, P-I(n) and H-I(n) are respectively the generalized Hilbert Samuel polynomial and the generalized Hilbert Samuel function. We then use it to establish formulas to compute the generalized Hilbert coefficients of I. As an application, we extend Northcott's inequality to non-m-primary ideals. Furthermore, when equality holds, we prove that the ideal I enjoys nice properties. Indeed, if this is the case, then the reduction number of I is at most one and the associated graded ring of I is Cohen-Macaulay. We also recover results of G. Colome-Nin, C. Polini, B. Ulrich and Y. Xie on the positivity of the generalized first Hilbert coefficient j(1)(I). Our work extends that of S. Huckaba, C. Huneke and A. Ooishi to ideals that are not necessarily m-primary. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free