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Distributive Lattices, Bipartite Graphs and Alexander Duality

Juergen Herzog·Takayuki Hibi
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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.

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Juergen Herzog, Takayuki Hibi. 2003-07-17. Distributive Lattices, Bipartite Graphs and Alexander Duality. https://arxiv.org/abs/math/0307235

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