### Abstract

Let A = K[x_{1},⋯,x_{n}] denote the polynomial ring in n variables over a field K with each deg X_{1} = 1. Let I be a homogeneous ideal of A with I ≠ A and H_{a}/i the Hilbert function of the quotient algebra A/I .Given a numerical function H : ℕ → ℕ satisfying H = H_{a/i} for some homogeneous ideal I of A,we write A _{h} for the set of those integers 0 ≤ r ≤ n such that there exists a homogeneous ideal I of A with H_{a/i} = H and with depth A/I = r. It will be proved that one has either Ah = {0,1,⋯,b} for some 0 ≤ b ≤ n or A|_{H}

Original language | English |
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Pages (from-to) | 1533-1538 |

Number of pages | 6 |

Journal | Proceedings of the American Mathematical Society |

Volume | 136 |

Issue number | 5 |

DOIs | |

Publication status | Published - 2008 May 1 |

Externally published | Yes |

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

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## Cite this

Murai, S., & Hibi, T. (2008). The depth of an ideal with a given hilbert function.

*Proceedings of the American Mathematical Society*,*136*(5), 1533-1538. https://doi.org/10.1090/S0002-9939-08-09067-9