Distributive Lattices, Bipartite Graphs and Alexander Duality
Abstract
A certain squarefree monomial ideal
H
P
arising from a finite partially ordered set
P
will be studied from viewpoints of both commutative algebra 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.