Cohen-Macaulay, shellable and unmixed clutters with a perfect matching of König type

• Autores: Susan Morey, Enrique Reyes, Rafael H. Villarreal
• Localización: Journal of pure and applied algebra, ISSN 0022-4049, Vol. 212, Nº 7, 2008, págs. 1770-1786
• Idioma: inglés
• DOI: 10.1016/j.jpaa.2007.11.010
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• Resumen

  • Let be a clutter with a perfect matching e1,�,eg of König type and let be the Stanley�Reisner complex of the edge ideal of . If all c-minors of have a free vertex and is unmixed, we show that is pure shellable. We are able to describe, in combinatorial and algebraic terms, when is pure. If has no cycles of length 3 or 4, then it is shown that is pure if and only if is pure shellable (in this case ei has a free vertex for all i), and that is pure if and only if for any two edges f1,f2 of and for any ei, one has that f1neif2nei or f2neif1nei. It is also shown that this ordering condition implies that is pure shellable, without any assumption on the cycles of . Then we prove that complete admissible uniform clutters and their Alexander duals are unmixed. In addition, the edge ideals of complete admissible uniform clutters are facet ideals of shellable simplicial complexes, they are Cohen�Macaulay, and they have linear resolutions. Furthermore if is admissible and complete, then is unmixed. We characterize certain conditions that occur in a Cohen�Macaulay criterion for bipartite graphs of Herzog and Hibi, and extend some results of Faridi�on the structure of unmixed simplicial trees�to clutters with the König property without 3-cycles or 4-cycles.