Abstract
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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Herzog, J., Hibi, T. Distributive Lattices, Bipartite Graphs and Alexander Duality. J Algebr Comb 22, 289–302 (2005). https://doi.org/10.1007/s10801-005-4528-1
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DOI: https://doi.org/10.1007/s10801-005-4528-1