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

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• 作者：Davide Bolognini ^{bolognin@dima.unige.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：13D02 ; 13A02 ; 05E40 ; 05E45
• 刊名：Journal of Pure and Applied Algebra
• 出版年：2016
• 出版时间：September 2016
• 年：2016
• 卷：220
• 期：9
• 页码：3102-3118
• 全文大小：484 K

文摘

A natural extension of bipartite graphs are d  -partite clutters, where d≥2$d\ge 2$ is an integer. For a poset P, Ene, Herzog and Mohammadi introduced the d  -partite clutter C_P,d${\mathcal{C}}_{P,d}$ of multichains of length d in P  , showing that it is Cohen–Macaulay. We prove that the cover ideal of C_P,d${\mathcal{C}}_{P,d}$ admits an x_i$x_i$-splitting, determining a recursive formula for its Betti numbers and generalizing a result of Francisco, Hà 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.