# The depth of an ideal with a given hilbert function

Satoshi Murai, Takayuki Hibi

Research output: Contribution to journalArticle

8 Citations (Scopus)

### Abstract

Let A = K[x1,⋯,xn] denote the polynomial ring in n variables over a field K with each deg X1 = 1. Let I be a homogeneous ideal of A with I ≠ A and Ha/i the Hilbert function of the quotient algebra A/I .Given a numerical function H : ℕ → ℕ satisfying H = Ha/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 Ha/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 1533-1538 6 Proceedings of the American Mathematical Society 136 5 https://doi.org/10.1090/S0002-9939-08-09067-9 Published - 2008 May 1 Yes

Hilbert Function
Algebra
H-function
Polynomials
Polynomial ring
Quotient
Denote
Integer

### ASJC Scopus subject areas

• Mathematics(all)
• Applied Mathematics

### Cite this

The depth of an ideal with a given hilbert function. / Murai, Satoshi; Hibi, Takayuki.

In: Proceedings of the American Mathematical Society, Vol. 136, No. 5, 01.05.2008, p. 1533-1538.

Research output: Contribution to journalArticle

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