### 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) |
---|---|

Pages (from-to) | 177-200 |

Number of pages | 24 |

Journal | Journal of Algebra |

Volume | 461 |

DOIs | |

State | Published - Sep 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

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

}

*Journal of Algebra*, vol. 461, pp. 177-200. https://doi.org/10.1016/j.jalgebra.2016.05.009

**Generalized Hilbert coefficients and Northcott's inequality.** / Xie, Yu.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Generalized Hilbert coefficients and Northcott's inequality

AU - Xie, Yu

PY - 2016/9/1

Y1 - 2016/9/1

N2 - 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 Gd condition and the weak Artin-Nagata property ANd-2-. We provide a formula relating the length λ(In+1/JIn) to the difference PI(n)-HI(n), where J is a general minimal reduction of I, PI(n) and HI(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 j1(I). Our work extends that of S. Huckaba, C. Huneke and A. Ooishi to ideals that are not necessarily m-primary.

AB - 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 Gd condition and the weak Artin-Nagata property ANd-2-. We provide a formula relating the length λ(In+1/JIn) to the difference PI(n)-HI(n), where J is a general minimal reduction of I, PI(n) and HI(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 j1(I). Our work extends that of S. Huckaba, C. Huneke and A. Ooishi to ideals that are not necessarily m-primary.

UR - http://www.scopus.com/inward/record.url?scp=84971207681&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84971207681&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2016.05.009

DO - 10.1016/j.jalgebra.2016.05.009

M3 - Article

AN - SCOPUS:84971207681

VL - 461

SP - 177

EP - 200

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -