Sequentially Cohen-Macaulay edge ideals

Authors:
Christopher A. Francisco and Adam Van Tuyl

Journal:
Proc. Amer. Math. Soc. **135** (2007), 2327-2337

MSC (2000):
Primary 13F55, 13D02, 05C38, 05C75

Published electronically:
March 21, 2007

MathSciNet review:
2302553

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a simple undirected graph on vertices, and let denote its associated edge ideal. We show that all chordal graphs are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of is componentwise linear. Our result complements Faridi's theorem that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay and implies Herzog, Hibi, and Zheng's theorem that a chordal graph is Cohen-Macaulay if and only if its edge ideal is unmixed. We also characterize the sequentially Cohen-Macaulay cycles and produce some examples of nonchordal sequentially Cohen-Macaulay graphs.

**1.**CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra, Available at`http://cocoa.dima.unige.it`**2.**Art M. Duval,*Algebraic shifting and sequentially Cohen-Macaulay simplicial complexes*, Electron. J. Combin.**3**(1996), no. 1, Research Paper 21, approx. 14 pp. (electronic). MR**1399398****3.**John A. Eagon and Victor Reiner,*Resolutions of Stanley-Reisner rings and Alexander duality*, J. Pure Appl. Algebra**130**(1998), no. 3, 265–275. MR**1633767**, 10.1016/S0022-4049(97)00097-2**4.**Sara Faridi,*Simplicial trees are sequentially Cohen-Macaulay*, J. Pure Appl. Algebra**190**(2004), no. 1-3, 121–136. MR**2043324**, 10.1016/j.jpaa.2003.11.014**5.**Sara Faridi,*Monomial ideals via square-free monomial ideals*, Commutative algebra, Lect. Notes Pure Appl. Math., vol. 244, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 85–114. MR**2184792**, 10.1201/9781420028324.ch8**6.**C. A. Francisco and H. Tài Hà, Whiskers and Sequentially Cohen-Macaulay graphs. (2006) Preprint.`arXiv:math.AC/0605487`.**7.**C. A. Francisco and A. Van Tuyl, Some families of componentwise linear monomial ideals. To appear,*Nagoya Math. J.***8.**D. R. Grayson and M. E. Stillman,*Macaulay 2, a software system for research in algebraic geometry*.

`http://www.math.uiuc.edu/Macaulay2/`

.**9.**Jürgen Herzog and Takayuki Hibi,*Componentwise linear ideals*, Nagoya Math. J.**153**(1999), 141–153. MR**1684555****10.**Jürgen Herzog and Takayuki Hibi,*Cohen-Macaulay polymatroidal ideals*, European J. Combin.**27**(2006), no. 4, 513–517. MR**2215212**, 10.1016/j.ejc.2005.01.004**11.**J. Herzog, T. Hibi, and X. Zheng, Cohen-Macaulay chordal graphs.*J. Combin. Theory Ser. A***113**(2006), no. 5, 911-916.**12.**Jürgen Herzog and Yukihide Takayama,*Resolutions by mapping cones*, Homology Homotopy Appl.**4**(2002), no. 2, 277–294. The Roos Festschrift volume, 2. MR**1918513****13.**Ezra Miller and Bernd Sturmfels,*Combinatorial commutative algebra*, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR**2110098****14.**Joseph J. Rotman,*An introduction to algebraic topology*, Graduate Texts in Mathematics, vol. 119, Springer-Verlag, New York, 1988. MR**957919****15.**Richard P. Stanley,*Combinatorics and commutative algebra*, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR**1453579****16.**Rafael H. Villarreal,*Monomial algebras*, Monographs and Textbooks in Pure and Applied Mathematics, vol. 238, Marcel Dekker, Inc., New York, 2001. MR**1800904**

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Additional Information

**Christopher A. Francisco**

Affiliation:
Department of Mathematics, Mathematical Sciences Building, University of Missouri, Columbia, Missouri 65203

Email:
chrisf@math.missouri.edu

**Adam Van Tuyl**

Affiliation:
Department of Mathematical Sciences, Lakehead University, Thunder Bay, ON P7B 5E1, Canada

Email:
avantuyl@sleet.lakeheadu.ca

DOI:
https://doi.org/10.1090/S0002-9939-07-08841-7

Keywords:
Componentwise linear,
sequentially Cohen-Macaulay,
edge ideals,
chordal graphs

Received by editor(s):
November 1, 2005

Received by editor(s) in revised form:
April 6, 2006

Published electronically:
March 21, 2007

Communicated by:
Michael Stillman

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.