### Abstract

Let M be a finite module, and let I be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of I on M using the zeroth local cohomology functor. We show that our definition reconciliates with that of Ciupercǎ. By generalizing Singh's formula (which holds in the case of λ(M/IM) < ∞), we prove that the generalized Hilbert coefficients j{fraktur}_{0},., j{fraktur}_{d-2} are preserved under a general hyperplane section, where d = dim M. We also keep track of the behavior of j{fraktur}_{d-1}. Then we apply these results to study the generalized Hilbert function for ideals that have minimal j-multiplicity or almost minimal j-multiplicity. We provide counterexamples to show that the generalized Hilbert series of ideals having minimal or almost minimal j-multiplicity does not have the 'expected' shape described in the case where λ(M/IM) < ∞. Finally, we give a sufficient condition such that the generalized Hilbert series has the desired shape.

Original language | English (US) |
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Pages (from-to) | 2411-2427 |

Number of pages | 17 |

Journal | Communications in Algebra |

Volume | 42 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2014 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Communications in Algebra*,

*42*(6), 2411-2427. https://doi.org/10.1080/00927872.2012.756884

}

*Communications in Algebra*, vol. 42, no. 6, pp. 2411-2427. https://doi.org/10.1080/00927872.2012.756884

**Generalized Hilbert Functions.** / Polini, Claudia; Xie, Yu.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Generalized Hilbert Functions

AU - Polini, Claudia

AU - Xie, Yu

PY - 2014/6/1

Y1 - 2014/6/1

N2 - Let M be a finite module, and let I be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of I on M using the zeroth local cohomology functor. We show that our definition reconciliates with that of Ciupercǎ. By generalizing Singh's formula (which holds in the case of λ(M/IM) < ∞), we prove that the generalized Hilbert coefficients j{fraktur}0,., j{fraktur}d-2 are preserved under a general hyperplane section, where d = dim M. We also keep track of the behavior of j{fraktur}d-1. Then we apply these results to study the generalized Hilbert function for ideals that have minimal j-multiplicity or almost minimal j-multiplicity. We provide counterexamples to show that the generalized Hilbert series of ideals having minimal or almost minimal j-multiplicity does not have the 'expected' shape described in the case where λ(M/IM) < ∞. Finally, we give a sufficient condition such that the generalized Hilbert series has the desired shape.

AB - Let M be a finite module, and let I be an arbitrary ideal over a Noetherian local ring. We define the generalized Hilbert function of I on M using the zeroth local cohomology functor. We show that our definition reconciliates with that of Ciupercǎ. By generalizing Singh's formula (which holds in the case of λ(M/IM) < ∞), we prove that the generalized Hilbert coefficients j{fraktur}0,., j{fraktur}d-2 are preserved under a general hyperplane section, where d = dim M. We also keep track of the behavior of j{fraktur}d-1. Then we apply these results to study the generalized Hilbert function for ideals that have minimal j-multiplicity or almost minimal j-multiplicity. We provide counterexamples to show that the generalized Hilbert series of ideals having minimal or almost minimal j-multiplicity does not have the 'expected' shape described in the case where λ(M/IM) < ∞. Finally, we give a sufficient condition such that the generalized Hilbert series has the desired shape.

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U2 - 10.1080/00927872.2012.756884

DO - 10.1080/00927872.2012.756884

M3 - Article

AN - SCOPUS:84893443348

VL - 42

SP - 2411

EP - 2427

JO - Communications in Algebra

JF - Communications in Algebra

SN - 0092-7872

IS - 6

ER -