Recursive Betti numbers for Cohen-Macaulay d-partite clutters arising from posets

摘要

A natural extension of bipartite graphs are d-partite clutters, where d >= 2 is an integer. For a poset P, Ene, Herzog and Mohammadi introduced the d-partite clutter C-P,C-d of multichains of length d in P, showing that it is Cohen-Macaulay. We prove that the cover ideal of C-P,C-d admits an x(i)-splitting, determining a recursive formula for its Betti numbers and generalizing a result of Francisco, Ha and Van Tuyl on the cover ideal of Cohen-Macaulay bipartite graphs. Moreover we prove a Betti splitting result for the Alexander dual of a Cohen Macaulay simplicial complex. Interesting examples are given, in particular the first example of ideal that does not admit Betti splitting in any characteristic. (C) 2016 Elsevier B.V. All rights reserved.