Sequentially Cohen-Macaulay edge ideals

Francisco, Christopher; Van Tuyl, Adam

doi:10.1090/S0002-9939-07-08841-7

Sequentially Cohen-Macaulay edge ideals
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by Christopher A. Francisco and Adam Van Tuyl PDF

Proc. Amer. Math. Soc. 135 (2007), 2327-2337 Request permission

Abstract:

Let $G$ be a simple undirected graph on $n$ vertices, and let $\mathcal I(G) \subseteq R = k[x_1,\ldots ,x_n]$ denote its associated edge ideal. We show that all chordal graphs $G$ are sequentially Cohen-Macaulay; our proof depends upon showing that the Alexander dual of $\mathcal I(G)$ 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.

References

CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra, Available at http://cocoa.dima.unige.it
Art M. Duval, Algebraic shifting and sequentially Cohen-Macaulay simplicial complexes, Electron. J. Combin. 3 (1996), no. 1, Research Paper 21, approx. 14. MR 1399398
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, DOI 10.1016/S0022-4049(97)00097-2
Sara Faridi, Simplicial trees are sequentially Cohen-Macaulay, J. Pure Appl. Algebra 190 (2004), no. 1-3, 121–136. MR 2043324, DOI 10.1016/j.jpaa.2003.11.014
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, DOI 10.1201/9781420028324.ch8
C. A. Francisco and H. Tài Hà, Whiskers and Sequentially Cohen-Macaulay graphs. (2006) Preprint. arXiv:math.AC/0605487.
C. A. Francisco and A. Van Tuyl, Some families of componentwise linear monomial ideals. To appear, Nagoya Math. J.
D. R. Grayson and M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/.
Jürgen Herzog and Takayuki Hibi, Componentwise linear ideals, Nagoya Math. J. 153 (1999), 141–153. MR 1684555, DOI 10.1017/S0027763000006930
Jürgen Herzog and Takayuki Hibi, Cohen-Macaulay polymatroidal ideals, European J. Combin. 27 (2006), no. 4, 513–517. MR 2215212, DOI 10.1016/j.ejc.2005.01.004
J. Herzog, T. Hibi, and X. Zheng, Cohen-Macaulay chordal graphs. J. Combin. Theory Ser. A 113 (2006), no. 5, 911–916.
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, DOI 10.4310/hha.2002.v4.n2.a13
Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227, Springer-Verlag, New York, 2005. MR 2110098
Joseph J. Rotman, An introduction to algebraic topology, Graduate Texts in Mathematics, vol. 119, Springer-Verlag, New York, 1988. MR 957919, DOI 10.1007/978-1-4612-4576-6
Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579
Rafael H. Villarreal, Monomial algebras, Monographs and Textbooks in Pure and Applied Mathematics, vol. 238, Marcel Dekker, Inc., New York, 2001. MR 1800904