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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
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References:
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