Distributive Lattices, Bipartite Graphs and Alexander Duality

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 whose Stanley–Reisner ideal coincides with HP is Cohen–Macaulay, all the Cohen–Macaulay bipartite graphs will be classified.