Journal of Algebraic Combinatorics

, Volume 22, Issue 3, pp 289–302 | Cite as

Distributive Lattices, Bipartite Graphs and Alexander Duality

Article

Abstract

A certain squarefree monomial ideal HP arising from a finite partially ordered set P will be studied from viewpoints of both commutative algbera and combinatorics. First, it is proved that the defining ideal of the Rees algebra of HP possesses a quadratic Gröbner basis. Thus in particular all powers of HP have linear resolutions. Second, the minimal free graded resolution of HP will be constructed explicitly and a combinatorial formula to compute the Betti numbers of HP will be presented. Third, by using the fact that the Alexander dual of the simplicial complex Δ whose Stanley–Reisner ideal coincides with HP is Cohen–Macaulay, all the Cohen–Macaulay bipartite graphs will be classified.

References

  1. 1.
    D. Bayer and B. Sturmfels, “Cellular resolutions of monomial modules,” J. Reine Angew. Math. 502 (1998), 123–140.MathSciNetGoogle Scholar
  2. 2.
    S. Blum, “Subalgebras of bigraded Koszul algebras,” J. Algebra 242 (2001), 795–809.CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    W. Bruns and J. Herzog, Cohen–Macaulay Rings, Revised Edition, Cambridge University Press, 1996.Google Scholar
  4. 4.
    J. Eagon and V. Reiner, “Resolutions of Stanley–Reisner rings and Alexander duality,” J. Pure Appl. Algebra 130 (1998), 265–275.CrossRefMathSciNetGoogle Scholar
  5. 5.
    D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer–Verlag, New York, NY, 1995.Google Scholar
  6. 6.
    R. Fröberg, Koszul algebras, “Advances in commutative ring theory” in D.E. Dobbs, M. Fontana and S.-E. Kabbaj (Eds.), Lecture Notes in Pure and Appl. Math., Vol. 205, Dekker, New York, NY, 1999, pp. 337–350.Google Scholar
  7. 7.
    J. Herzog, T. Hibi, and X. Zheng, “Dirac's theorem on chordal graphs and Alexander duality,” European J. Comb. 25(7) (2004), 826–838.MathSciNetGoogle Scholar
  8. 8.
    J. Herzog, T. Hibi, and X. Zheng, “The monomial ideal of a finite meet semi-lattice,” to appear in Trans. AMS.Google Scholar
  9. 9.
    T. Hibi, “Distributive lattices, affine semigroup rings and algebras with straightening laws,” in Commutative Algebra and Combinatorics, Advanced Studies in Pure Math., M. Nagata and H. Matsumura, (Eds.), Vol. 11, North–Holland, Amsterdam, 1987, pp. 93–109.Google Scholar
  10. 10.
    T. Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw, Glebe, N.S.W., Australia, 1992.Google Scholar
  11. 11.
    C. Peskine and L. Szpiro, “Syzygies and multiplicities,” C.R. Acad. Sci. Paris. Sér. A 278 (1974), 1421–1424.MathSciNetGoogle Scholar
  12. 12.
    R.P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole, Monterey, CA, 1986.Google Scholar
  13. 13.
    R.P. Stanley, Combinatorics and Commutative Algebra, Second Edition, Birkhäuser, Boston, MA, 1996.Google Scholar
  14. 14.
    B. Sturmfels, “Gröbner Bases and Convex Polytopes,” Amer. Math. Soc., Providence, RI, 1995.Google Scholar
  15. 15.
    R.H. Villareal, Monomial Algebras, Dekker, New York, NY, 2001.Google Scholar

Copyright information

© Springer Science + Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Fachbereich Mathematik und InformatikUniversität Duisburg-EssenEssenGermany
  2. 2.Department of Pure and Applied MathematicsGraduate School of Information Science and Technology, Osaka UniversityToyonaka, OsakaJapan

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