### Abstract

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 ANd-2-. We provide a formula relating the length λ(I^{n+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. Colomé-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.

Original language | English (US) |
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Pages (from-to) | 177-200 |

Number of pages | 24 |

Journal | Journal of Algebra |

Volume | 461 |

DOIs | |

State | Published - Sep 1 2016 |

### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

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## Cite this

*Journal of Algebra*,

*461*, 177-200. https://doi.org/10.1016/j.jalgebra.2016.05.009