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 | References | Similar Articles | Additional Information

Abstract: Let denote the polynomial ring in variables over a field with each . Let be a homogeneous ideal of with and the Hilbert function of the quotient algebra . Given a numerical function satisfying for some homogeneous ideal of , we write for the set of those integers such that there exists a homogeneous ideal of with and with . It will be proved that one has either for some or .

**1.**Eric Babson, Isabella Novik, and Rekha Thomas,*Reverse lexicographic and lexicographic shifting*, J. Algebraic Combin.**23**(2006), no. 2, 107–123. MR**2223682**, 10.1007/s10801-006-6919-3**2.**Anna Maria Bigatti,*Upper bounds for the Betti numbers of a given Hilbert function*, Comm. Algebra**21**(1993), no. 7, 2317–2334. MR**1218500**, 10.1080/00927879308824679**3.**Winfried Bruns and Jürgen Herzog,*Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR**1251956****4.**Aldo Conca,*Koszul homology and extremal properties of Gin and Lex*, Trans. Amer. Math. Soc.**356**(2004), no. 7, 2945–2961. MR**2052603**, 10.1090/S0002-9947-03-03393-2**5.**Shalom Eliahou and Michel Kervaire,*Minimal resolutions of some monomial ideals*, J. Algebra**129**(1990), no. 1, 1–25. MR**1037391**, 10.1016/0021-8693(90)90237-I**6.**Jürgen Herzog,*Generic initial ideals and graded Betti numbers*, Computational commutative algebra and combinatorics (Osaka, 1999) Adv. Stud. Pure Math., vol. 33, Math. Soc. Japan, Tokyo, 2002, pp. 75–120. MR**1890097****7.**Heather A. Hulett,*Maximum Betti numbers of homogeneous ideals with a given Hilbert function*, Comm. Algebra**21**(1993), no. 7, 2335–2350. MR**1218501**, 10.1080/00927879308824680**8.**Keith Pardue,*Deformation classes of graded modules and maximal Betti numbers*, Illinois J. Math.**40**(1996), no. 4, 564–585. MR**1415019**

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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

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.