Distributive Lattices, Bipartite Graphs and Alexander Duality
Jürgen Herzog1
and Takayuki Hibi2
1Universität Duisburg-Essen Fachbereich Mathematik und Informatik Campus Essen 45117 Essen Germany Campus Essen 45117 Essen Germany
2Graduate School of Information Science and Technology, Osaka University Department of Pure and Applied Mathematics Toyonaka, Osaka 560-0043 Japan Toyonaka, Osaka 560-0043 Japan
2Graduate School of Information Science and Technology, Osaka University Department of Pure and Applied Mathematics Toyonaka, Osaka 560-0043 Japan Toyonaka, Osaka 560-0043 Japan
DOI: 10.1007/s10801-005-4528-1
Abstract
A certain squarefree monomial ideal H P 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 H P possesses a quadratic Gröbner basis. Thus in particular all powers of H P have linear resolutions. Second, the minimal free graded resolution of H P will be constructed explicitly and a combinatorial formula to compute the Betti numbers of H P will be presented. Third, by using the fact that the Alexander dual of the simplicial complex Δ whose Stanley-Reisner ideal coincides with H P is Cohen-Macaulay, all the Cohen-Macaulay bipartite graphs will be classified.
Pages: 289–302
Full Text: PDF
References
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2. S. Blum, “Subalgebras of bigraded Koszul algebras,” J. Algebra 242 (2001), 795-809.
3. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Revised Edition, Cambridge University Press, 1996.
4. J. Eagon and V. Reiner, “Resolutions of Stanley-Reisner rings and Alexander duality,” J. Pure Appl. Algebra 130 (1998), 265-275.
5. D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, New York, NY, 1995.
6. R. Fr\ddot oberg, 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.
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.
8. J. Herzog, T. Hibi, and X. Zheng, “The monomial ideal of a finite meet semi-lattice,” to appear in Trans. AMS.
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.
10. T. Hibi, Algebraic Combinatorics on Convex Polytopes, Carslaw, Glebe, N.S.W., Australia, 1992.
11. C. Peskine and L. Szpiro, “Syzygies and multiplicities,” C.R. Acad. Sci. Paris. Sér. A 278 (1974), 1421-1424.
12. R.P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole, Monterey, CA, 1986.
13. R.P. Stanley, Combinatorics and Commutative Algebra, Second Edition, Birkh\ddot auser, Boston, MA, 1996.
14. B. Sturmfels, “Gr\ddot obner Bases and Convex Polytopes,” Amer. Math. Soc., Providence, RI, 1995.
15. R.H. Villareal, Monomial Algebras, Dekker, New York, NY, 2001.