The depth of an ideal with a given Hilbert function
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- by Satoshi Murai and Takayuki Hibi PDF
- Proc. Amer. Math. Soc. 136 (2008), 1533-1538 Request permission
Abstract:
Let $A = K[x_1, \ldots , x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\operatorname {deg} x_i = 1$. Let $I$ be a homogeneous ideal of $A$ with $I \neq A$ and $H_{A/I}$ the Hilbert function of the quotient algebra $A / I$. Given a numerical function $H : {\mathbb {N}} \to {\mathbb {N}}$ satisfying $H=H_{A/I}$ for some homogeneous ideal $I$ of $A$, we write $\mathcal {A} _H$ for the set of those integers $0 \leq r \leq n$ such that there exists a homogeneous ideal $I$ of $A$ with $H_{A/I} = H$ and with $\operatorname {depth} A / I = r$. It will be proved that one has either $\mathcal {A}_H = \{ 0, 1, \ldots , b \}$ for some $0 \leq b \leq n$ or $|{\mathcal {A}}_H| = 1$.References
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Additional Information
- Satoshi Murai
- Affiliation: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka, 560-0043, Japan
- MR Author ID: 800440
- Email: s-murai@ist.osaka-u.ac.jp
- Takayuki Hibi
- Affiliation: Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Toyonaka, Osaka, 560-0043, Japan
- MR Author ID: 219759
- Email: hibi@math.sci.osaka-u.ac.jp
- Received by editor(s): August 9, 2006
- Received by editor(s) in revised form: December 5, 2006
- Published electronically: January 17, 2008
- Additional Notes: The first author is supported by JSPS Research Fellowships for Young Scientists
- Communicated by: Bernd Ulrich
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 136 (2008), 1533-1538
- MSC (2000): Primary 13C15; Secondary 13D40
- DOI: https://doi.org/10.1090/S0002-9939-08-09067-9
- MathSciNet review: 2373580