Sequentially CohenMacaulay edge ideals
Authors:
Christopher A. Francisco and Adam Van Tuyl
Journal:
Proc. Amer. Math. Soc. 135 (2007), 23272337
MSC (2000):
Primary 13F55, 13D02, 05C38, 05C75
Published electronically:
March 21, 2007
MathSciNet review:
2302553
Fulltext PDF Free Access
Abstract 
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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 CohenMacaulay; 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 CohenMacaulay and implies Herzog, Hibi, and Zheng's theorem that a chordal graph is CohenMacaulay if and only if its edge ideal is unmixed. We also characterize the sequentially CohenMacaulay cycles and produce some examples of nonchordal sequentially CohenMacaulay 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 CohenMacaulay
simplicial complexes, Electron. J. Combin. 3 (1996),
no. 1, Research Paper 21, approx.\ 14 pp.\ (electronic). MR 1399398
(98b:06009)
 3.
John
A. Eagon and Victor
Reiner, Resolutions of StanleyReisner rings and Alexander
duality, J. Pure Appl. Algebra 130 (1998),
no. 3, 265–275. MR 1633767
(99h:13017), http://dx.doi.org/10.1016/S00224049(97)000972
 4.
Sara
Faridi, Simplicial trees are sequentially CohenMacaulay, J.
Pure Appl. Algebra 190 (2004), no. 13,
121–136. MR 2043324
(2004m:13058), http://dx.doi.org/10.1016/j.jpaa.2003.11.014
 5.
Sara
Faridi, Monomial ideals via squarefree monomial ideals,
Commutative algebra, Lect. Notes Pure Appl. Math., vol. 244, Chapman
& Hall/CRC, Boca Raton, FL, 2006, pp. 85–114. MR 2184792
(2006i:13038), http://dx.doi.org/10.1201/9781420028324.ch8
 6.
C. A. Francisco and H. Tài Hà, Whiskers and Sequentially CohenMacaulay 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
(2000i:13019)
 10.
Jürgen
Herzog and Takayuki
Hibi, CohenMacaulay polymatroidal ideals, European J. Combin.
27 (2006), no. 4, 513–517. MR 2215212
(2007e:13017), http://dx.doi.org/10.1016/j.ejc.2005.01.004
 11.
J. Herzog, T. Hibi, and X. Zheng, CohenMacaulay chordal graphs. J. Combin. Theory Ser. A 113 (2006), no. 5, 911916.
 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
(2003k:13014)
 13.
Ezra
Miller and Bernd
Sturmfels, Combinatorial commutative algebra, Graduate Texts
in Mathematics, vol. 227, SpringerVerlag, New York, 2005. MR 2110098
(2006d:13001)
 14.
Joseph
J. Rotman, An introduction to algebraic topology, Graduate
Texts in Mathematics, vol. 119, SpringerVerlag, New York, 1988. MR 957919
(90e:55001)
 15.
Richard
P. Stanley, Combinatorics and commutative algebra, 2nd ed.,
Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc.,
Boston, MA, 1996. MR 1453579
(98h:05001)
 16.
Rafael
H. Villarreal, Monomial algebras, Monographs and Textbooks in
Pure and Applied Mathematics, vol. 238, Marcel Dekker, Inc., New York,
2001. MR
1800904 (2002c:13001)
 1.
 CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra, Available at http://cocoa.dima.unige.it
 2.
 A. M. Duval, Algebraic shifting and sequentially CohenMacaulay simplicial complexes. Electron. J. Combin. 3 (1996), no. 1, Research Paper 21, approx. 14 pp. (electronic). MR 1399398 (98b:06009)
 3.
 J. Eagon and V. Reiner, Resolutions of StanleyReisner rings and Alexander duality. J. Pure Appl. Algebra 130 (1998), no. 3, 265275. MR 1633767 (99h:13017)
 4.
 S. Faridi, Simplicial trees are sequentially CohenMacaulay. J. Pure Appl. Algebra 190 (2003), 121136. MR 2043324 (2004m:13058)
 5.
 S. Faridi, Monomial ideals via squarefree monomial ideals, Lecture Notes in Pure and Applied Mathematics 244 (2005) 85114. MR 2184792 (2006i:13038)
 6.
 C. A. Francisco and H. Tài Hà, Whiskers and Sequentially CohenMacaulay 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.
 J. Herzog and T. Hibi, Componentwise linear ideals. Nagoya Math. J. 153 (1999), 141153. MR 1684555 (2000i:13019)
 10.
 J. Herzog and T. Hibi, CohenMacaulay polymatroidal ideals. European J. Combin. 27 (2006), no. 4, 513517. MR 2215212
 11.
 J. Herzog, T. Hibi, and X. Zheng, CohenMacaulay chordal graphs. J. Combin. Theory Ser. A 113 (2006), no. 5, 911916.
 12.
 J. Herzog and Y. Takayama, Resolutions by mapping cones. The Roos Festschrift volume, 2. Homology Homotopy Appl. 4 (2002), no. 2, part 2, 277294 (electronic). MR 1918513 (2003k:13014)
 13.
 E. Miller and B. Sturmfels, Combinatorial commutative algebra. Springer, 2005. MR 2110098 (2006d:13001)
 14.
 J. Rotman, An introduction to algebraic topology. SpringerVerlag, 1988. MR 0957919 (90e:55001)
 15.
 R. P. Stanley, Combinatorics and commutative algebra. Second edition. Progress in Mathematics 41. Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579 (98h:05001)
 16.
 R. Villarreal, Monomial Algebras. Marcel Dekker, 2001. MR 1800904 (2002c:13001)
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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:
http://dx.doi.org/10.1090/S0002993907088417
PII:
S 00029939(07)088417
Keywords:
Componentwise linear,
sequentially CohenMacaulay,
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.
