Previous |  Up |  Next

Article

Full entry | PDF   (0.3 MB)
Keywords:
Betti numbers; chordal hypergraphs; connectivity; homologically connected hypergraphs; hypercycles; line hypergraphs; shellability
Summary:
We consider Stanley-Reisner rings $k[x_1,\ldots ,x_n]/I(\mathcal {H})$ where $I(\mathcal {H})$ is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize some known results about chordal graphs and study a weak form of shellability.
References:
[1] Berge, C.: Hypergraphs: Combinatorics of finite sets. {North-Holland} (1989). MR 1013569 | Zbl 0674.05001
[2] Bruns, W., Herzog, J.: Cohen-Macaulay rings, revised ed. Cambridge University Press (1998). MR 1251956
[3] Dochstermann, A., Engström, A.: Algebraic properties of edge ideals via combinatorial topology. arXiv:0810.4120 (2008). MR 2992613
[4] Eagon, J. A., Reiner, V.: Resolutions of Stanley-Reisner rings and Alexander Duality. J. Pure Appl. Algebra 130 (1998), 265-275. DOI 10.1016/S0022-4049(97)00097-2 | MR 1633767 | Zbl 0941.13016
[5] Eisenbud, D.: Commutative algebra with a view toward algebraic geometry. Graduate Texts in Mathematics 150, Springer (1995). MR 1322960 | Zbl 0819.13001
[6] Eliahou, S., Kervaire, M.: Minimal resolutions of some monomial ideals. J. Algebra 129 (1990), 1-25. DOI 10.1016/0021-8693(90)90237-I | MR 1037391 | Zbl 0701.13006
[7] Emtander, E.: Betti numbers of hypergraphs. Communications in algebra 37 (2009), 1545-1571. DOI 10.1080/00927870802098158 | MR 2526320 | Zbl 1191.13015
[8] Emtander, E.: A class of hypergraphs that generalizes chordal graphs. arXiv:0803.2150 (2008). MR 2603461
[9] Faridi, S.: The facet ideal of a simplicial complex. Manuscripta Math. 242 (2002), 92-108. MR 1935027 | Zbl 1005.13006
[10] Faridi, S.: Cohen-Macaulay Properties of Square-Free Monomial Ideals. J. Combin. Theory Ser. A 109 (2005), 299-329. DOI 10.1016/j.jcta.2004.09.005 | MR 2121028 | Zbl 1101.13015
[11] Francisco, C. A., Tuyl, A. Van: Sequentially cohen-macaulay edge ideals. Proc. Amer. Math. Soc. (2007), 2327-2337. MR 2302553
[12] Francisco, Christopher A., Hà, H. T., Tuyl, A. Van: Splittings of monomial ideals. arXiv:0807.2185 (2008). MR 2515396
[13] Fröberg, R.: Rings with monomial relations having linear resolutions. J. Pure Appl. Algebra 38 (1985), 235-241. DOI 10.1016/0022-4049(85)90011-8 | MR 0814179
[14] Fröberg, R.: On Stanley-Reisner rings. Topics in Algebra 26 (1990). MR 1171260
[15] Hà, H. T., Tuyl, A. Van: Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers. arXiv:math/0606539 (2006). MR 2375493
[16] Hà, H. T., Tuyl, A. Van: Splittable ideals and resolutions of monomial ideals. J. Algebra 309 (2007), 405-425. DOI 10.1016/j.jalgebra.2006.08.022 | MR 2301246
[17] Herzog, J., Hibi, T., Zheng, X.: Cohen-Macaulay chordal graphs. arXiv:math/0407375v1 (2004). MR 2231097
[18] Herzog, J., Hibi, T., Zheng, X.: Dirac's theorem on chordal graphs and Alexander duality. European Journal of Combinatorics 25 (2004). DOI 10.1016/j.ejc.2003.12.008 | MR 2083448 | Zbl 1062.05075
[19] Jacques, S.: Betti Numbers of Graph Ideals. Ph.D. thesis, University of Sheffield, 2004, arXiv:math/0410107.
[20] Miller, E., Sturmfels, B.: Combinatorial Commutative Algebra. Springer (2005). MR 2110098 | Zbl 1090.13001
[21] Morey, S., Reyes, E., Villarreal, R. H.: Cohen-Macaulay, Shellable and unmixed clutters with a perfect matching of König type. arXiv:0708.3111v3 (2007). MR 2400742
[22] Villarreal, R. H.: Cohen-Macaulay graphs. Manuscripta Math. 66 (1990), 277-293. DOI 10.1007/BF02568497 | MR 1031197 | Zbl 0737.13003
[23] Zheng, X.: Resolutions of facet ideals. Comm. Algebra 32 (2004), 2301-2324. DOI 10.1081/AGB-120037222 | MR 2100472 | Zbl 1089.13014

Partner of