### Abstract

In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to d, where d is a positive integer. In addition, we prove the following result which is a generalization of Bigatti, Hulett and Pardue's result: Let p ≥ 0 and d > 0 be integers. If the base field is a field of characteristic 0 and there is a graded ideal I whose projective dimension proj dim (I) is smaller than or equal to p and whose regularity reg (I) is smaller than or equal to d, then there exists a monomial ideal L having the maximal graded Betti numbers among graded ideals J which have the same Hilbert function as I and which satisfy proj dim (J) ≤ p and reg (J) ≤ d. We also prove the same fact for squarefree monomial ideals. The main methods for proofs are generic initial ideals and combinatorics on strongly stable ideals.

Original language | English |
---|---|

Pages (from-to) | 658-690 |

Number of pages | 33 |

Journal | Journal of Algebra |

Volume | 317 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2007 Nov 15 |

Externally published | Yes |

### Fingerprint

### Keywords

- Castelnuovo-Mumford regularity
- Generic initial ideals
- Graded Betti numbers
- Hilbert functions
- Lexsegment ideals

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Algebra*,

*317*(2), 658-690. https://doi.org/10.1016/j.jalgebra.2007.05.008

**Hilbert functions of d-regular ideals.** / Murai, Satoshi.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 317, no. 2, pp. 658-690. https://doi.org/10.1016/j.jalgebra.2007.05.008

}

TY - JOUR

T1 - Hilbert functions of d-regular ideals

AU - Murai, Satoshi

PY - 2007/11/15

Y1 - 2007/11/15

N2 - In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to d, where d is a positive integer. In addition, we prove the following result which is a generalization of Bigatti, Hulett and Pardue's result: Let p ≥ 0 and d > 0 be integers. If the base field is a field of characteristic 0 and there is a graded ideal I whose projective dimension proj dim (I) is smaller than or equal to p and whose regularity reg (I) is smaller than or equal to d, then there exists a monomial ideal L having the maximal graded Betti numbers among graded ideals J which have the same Hilbert function as I and which satisfy proj dim (J) ≤ p and reg (J) ≤ d. We also prove the same fact for squarefree monomial ideals. The main methods for proofs are generic initial ideals and combinatorics on strongly stable ideals.

AB - In the present paper, we characterize all possible Hilbert functions of graded ideals in a polynomial ring whose regularity is smaller than or equal to d, where d is a positive integer. In addition, we prove the following result which is a generalization of Bigatti, Hulett and Pardue's result: Let p ≥ 0 and d > 0 be integers. If the base field is a field of characteristic 0 and there is a graded ideal I whose projective dimension proj dim (I) is smaller than or equal to p and whose regularity reg (I) is smaller than or equal to d, then there exists a monomial ideal L having the maximal graded Betti numbers among graded ideals J which have the same Hilbert function as I and which satisfy proj dim (J) ≤ p and reg (J) ≤ d. We also prove the same fact for squarefree monomial ideals. The main methods for proofs are generic initial ideals and combinatorics on strongly stable ideals.

KW - Castelnuovo-Mumford regularity

KW - Generic initial ideals

KW - Graded Betti numbers

KW - Hilbert functions

KW - Lexsegment ideals

UR - http://www.scopus.com/inward/record.url?scp=35348952116&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=35348952116&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2007.05.008

DO - 10.1016/j.jalgebra.2007.05.008

M3 - Article

AN - SCOPUS:35348952116

VL - 317

SP - 658

EP - 690

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 2

ER -