# zbMATH — the first resource for mathematics

Distributive lattices, bipartite graphs and Alexander duality. (English) Zbl 1090.13017
In the paper under review, a certain squarefree monomial ideal $$H_{P}$$ arising from a finite partially ordered set $$P$$ is studied from the viewpoints of both commutative algebra and combinatorics. It is shown that the defining ideal of the Rees algebra $$R(H_{P})$$ of $$H_{P}$$ possesses a reduced Gröbner basis consisting of quadratic binomials whose initial monomials are squarefree and as a consequence one obtains that $$\text{R}(H_{P})$$ turns out to be normal and Koszul, and applying a result of S. Blum [J. Algebra 242, 795–809 (2001; Zbl 1042.13001)] it is shown that all powers of the ideal $$H_{P}$$ have linear resolutions. The minimal free graded resolution of $$H_{P}$$ is constructed explicitly and the Betti numbers $$\beta_{i}$$ of $$H_{P}$$ are given by the number of intervals of a finite distributive lattice $${L}$$ which are isomorphic to Boolean lattices of rank $$i$$. It is shown that the ideal $$H_{P}$$ is of height $$2$$ and a formula to compute the multiplicity of $$S/H_{P}$$ is given. In addition, by using the fact that the Alexander dual of the simplicial complex $$\Delta$$ whose Stanley-Reisner ideal coincides with $$H_{P}$$ is Cohen-Macaulay, an interesting classification of all the Cohen-Macaulay bipartite graphs is given.

##### MSC:
 13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes 13D02 Syzygies, resolutions, complexes and commutative rings 05C90 Applications of graph theory 06D05 Structure and representation theory of distributive lattices
Full Text:
##### References:
 [1] Bayer, D.; Sturmfels, B., Cellular resolutions of monomial modules, J. Reine Angew. Math., 502, 123-140, (1998) · Zbl 0909.13011 [2] Blum, S., Subalgebras of bigraded Koszul algebras, J. Algebra, 242, 795-809, (2001) · Zbl 1042.13001 [3] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Revised Edition, Cambridge University Press, 1996. [4] Eagon, J.; Reiner, V., Resolutions of Stanley-Reisner rings and Alexander duality, J. Pure Appl. Algebra, 130, 265-275, (1998) · Zbl 0941.13016 [5] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Springer-Verlag, New York, NY, 1995. [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. [7] Herzog, J.; Hibi, T.; Zheng, X., Dirac’s theorem on chordal graphs and Alexander duality, European J. Comb., 25, 826-838, (2004) [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. · Zbl 0772.52008 [11] Peskine, C.; Szpiro, L., Syzygies and multiplicities, C.R. Acad. Sci. Paris. Sér. A, 278, 1421-1424, (1974) · Zbl 0281.13004 [12] R.P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth & Brooks/Cole, Monterey, CA, 1986. [13] R.P. Stanley, Combinatorics and Commutative Algebra, Second Edition, Birkhäuser, Boston, MA, 1996. [14] B. Sturmfels, “Gröbner Bases and Convex Polytopes,” Amer. Math. Soc., Providence, RI, 1995. [15] R.H. Villareal, Monomial Algebras, Dekker, New York, NY, 2001.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.