Higher chordality: From graphs to complexes
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- by Karim A. Adiprasito, Eran Nevo and Jose A. Samper PDF
- Proc. Amer. Math. Soc. 144 (2016), 3317-3329 Request permission
Abstract:
We generalize the fundamental graph-theoretic notion of chordality for higher dimensional simplicial complexes by putting it into a proper context within homology theory. We generalize some of the classical results of graph chordality to this generality, including the fundamental relation to the Leray property and chordality theorems of Dirac.References
- K. A. Adiprasito and R. Sanyal, Relative Stanley–Reisner Theory and Upper Bound Theorems for Minkowski sums, preprint, arXiv:1405.7368.
- B. Benedetti and F. H. Lutz, The dunce hat and a minimal non-extendably collapsible 3-ball, Electronic Geometry Model No. 2013.10.001. Available online (http: //www.eg-models.de/2013.10.001) (2013).
- E. Connon and S. Faridi, Chorded complexes and a necessary condition for a monomial ideal to have a linear resolution, J. Combin. Theory Ser. A 120 (2013), no. 7, 1714–1731. MR 3092695, DOI 10.1016/j.jcta.2013.05.009
- E. Connon and S. Faridi, A criterion for a monomial ideal to have a linear resolution in characteristic $2$, math arXiv 1306.2857 (2013).
- Raul Cordovil, Manoel Lemos, and Cláudia Linhares Sales, Dirac’s theorem on simplicial matroids, Ann. Comb. 13 (2009), no. 1, 53–63. MR 2529719, DOI 10.1007/s00026-009-0012-2
- Michael W. Davis and Tadeusz Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2, 417–451. MR 1104531, DOI 10.1215/S0012-7094-91-06217-4
- G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961), 71–76. MR 130190, DOI 10.1007/BF02992776
- 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
- Eric Emtander, A class of hypergraphs that generalizes chordal graphs, Math. Scand. 106 (2010), no. 1, 50–66. MR 2603461, DOI 10.7146/math.scand.a-15124
- Ralf Fröberg, On Stanley-Reisner rings, Topics in algebra, Part 2 (Warsaw, 1988) Banach Center Publ., vol. 26, PWN, Warsaw, 1990, pp. 57–70. MR 1171260
- Mark L. Green, Generic initial ideals, Six lectures on commutative algebra (Bellaterra, 1996) Progr. Math., vol. 166, Birkhäuser, Basel, 1998, pp. 119–186. MR 1648665
- Huy Tài Hà and Adam Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebraic Combin. 27 (2008), no. 2, 215–245. MR 2375493, DOI 10.1007/s10801-007-0079-y
- M. Hachimori, Nonshellable but constructible 2-dimensional simplicial complex., Electronic Geometry Model (2003) (English).
- J. Herzog and H. Srinivasan, A note on the subadditivity problem for maximal shifts in free resolutions, math arXiv 1303.6214 (2013).
- Gil Kalai, Rigidity and the lower bound theorem. I, Invent. Math. 88 (1987), no. 1, 125–151. MR 877009, DOI 10.1007/BF01405094
- Gil Kalai and Roy Meshulam, Intersections of Leray complexes and regularity of monomial ideals, J. Combin. Theory Ser. A 113 (2006), no. 7, 1586–1592. MR 2259083, DOI 10.1016/j.jcta.2006.01.005
- C. W. Lee, P.L.-spheres, convex polytopes, and stress, Discrete Comput. Geom. 15 (1996), no. 4, 389–421. MR 1384883, DOI 10.1007/BF02711516
- C. G. Lekkerkerker and J. Ch. Boland, Representation of a finite graph by a set of intervals on the real line, Fund. Math. 51 (1962/63), 45–64. MR 139159, DOI 10.4064/fm-51-1-45-64
- Anda Olteanu, Constructible ideals, Comm. Algebra 37 (2009), no. 5, 1656–1669. MR 2526328, DOI 10.1080/00927870802210001
- Richard P. Stanley, Combinatorics and commutative algebra, 2nd ed., Progress in Mathematics, vol. 41, Birkhäuser Boston, Inc., Boston, MA, 1996. MR 1453579
- Tiong-Seng Tay, Neil White, and Walter Whiteley, Skeletal rigidity of simplicial complexes. I, European J. Combin. 16 (1995), no. 4, 381–403. MR 1337142, DOI 10.1016/0195-6698(95)90019-5
- Russ Woodroofe, Chordal and sequentially Cohen-Macaulay clutters, Electron. J. Combin. 18 (2011), no. 1, Paper 208, 20. MR 2853065
- E. C. Zeeman, On the dunce hat, Topology 2 (1964), 341–358. MR 156351, DOI 10.1016/0040-9383(63)90014-4
Additional Information
- Karim A. Adiprasito
- Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540 – and – Einstein Institute of Mathematics, University of Jerusalem, Jerusalem, Israel
- MR Author ID: 963585
- Email: adiprasito@math.huji.ac.il
- Eran Nevo
- Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel
- MR Author ID: 762118
- Email: nevo@math.huji.ac.il
- Jose A. Samper
- Affiliation: Department of Mathematics, University of Washington at Seattle, Seattle, Washington 98105
- Email: samper@math.washington.edu
- Received by editor(s): May 13, 2015
- Received by editor(s) in revised form: October 17, 2015
- Published electronically: February 3, 2016
- Additional Notes: The first author acknowledges support by an IPDE/EPDI postdoctoral fellowship, a Minerva postdoctoral fellowship of the Max Planck Society, and NSF Grant DMS 1128155
The research of the second author was partially supported by an ISF grant 805/11 - Communicated by: Irena Peeva
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 3317-3329
- MSC (2010): Primary 05Cxx, 05E45, 13F55
- DOI: https://doi.org/10.1090/proc/13002
- MathSciNet review: 3503700