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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The depth of an ideal with a given Hilbert function


Authors: Satoshi Murai and Takayuki Hibi
Journal: Proc. Amer. Math. Soc. 136 (2008), 1533-1538
MSC (2000): Primary 13C15; Secondary 13D40
Published electronically: January 17, 2008
MathSciNet review: 2373580
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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 $ \vert{\mathcal{A}}_H\vert = 1$.


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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
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
Email: hibi@math.sci.osaka-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09067-9
PII: S 0002-9939(08)09067-9
Keywords: Hilbert functions, depth, lexsegment ideals
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.