Distributive lattices, bipartite graphs and Alexander duality

Herzog, Jürgen; Hibi, Takayuki

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Distributive lattices, bipartite graphs and Alexander duality

Herzog, Jürgen ; Hibi, Takayuki

Journal of Algebraic Combinatorics, Tome 22 (2005) no. 3, pp. 289-302.

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Résumé

Summary: A certain squarefree monomial ideal

H_{P}

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

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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@article{JAC_2005__22_3_a4,
     author = {Herzog, J\"urgen and Hibi, Takayuki},
     title = {Distributive lattices, bipartite graphs and {Alexander} duality},
     journal = {Journal of Algebraic Combinatorics},
     pages = {289--302},
     publisher = {mathdoc},
     volume = {22},
     number = {3},
     year = {2005},
     language = {en},
     url = {https://geodesic-test.mathdoc.fr/item/JAC_2005__22_3_a4/}
}

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EP  - 302
VL  - 22
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%A Hibi, Takayuki
%T Distributive lattices, bipartite graphs and Alexander duality
%J Journal of Algebraic Combinatorics
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%V 22
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Herzog, Jürgen; Hibi, Takayuki. Distributive lattices, bipartite graphs and Alexander duality. Journal of Algebraic Combinatorics, Tome 22 (2005) no. 3, pp. 289-302. https://geodesic-test.mathdoc.fr/item/JAC_2005__22_3_a4/