A generalization of$k$-Cohen–Macaulay simplicial complexes

Hassan Haghighi Department of Mathematics, K. N. Toosi University of Technology Siamak Yassemi School of Mathematics, Statistics & Computer Science College of Science, University of Tehran Rahim Zaare-Nahandi School of Mathematics, Statistics & Computer Science College of Science, University of Tehran


Arkiv for Matematik, 50, (2), 279-290, 2010.6
For a positive integer$k$and a non-negative integer$t$, a class of simplicial complexes, to be denoted by$k$-CM_{$t$}, is introduced. This class generalizes two notions for simplicial complexes: being$k$-Cohen–Macaulay and$k$-Buchsbaum. In analogy with the Cohen–Macaulay and Buchsbaum complexes, we give some characterizations of CM_{$t$}(=1−CM_{$t$}) complexes, in terms of vanishing of some homologies of its links, and in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We give a result on the behavior of the CM_{$t$}property under the operation of join of two simplicial complexes. We show that a complex is$k$-CM_{$t$}if and only if the links of its non-empty faces are$k$-CM_{$t$−1}. We prove that for an integer$s$≤$d$, the ($d$−$s$−1)-skeleton of a ($d$−1)-dimensional$k$-CM_{$t$}complex is ($k$+$s$)-CM_{$t$}. This result generalizes Hibi’s result for Cohen–Macaulay complexes and Miyazaki’s result for Buchsbaum complexes.
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  title={A generalization of$k$-Cohen–Macaulay simplicial complexes},
  author={Hassan Haghighi, Siamak Yassemi, and Rahim Zaare-Nahandi},
  booktitle={Arkiv for Matematik},
Hassan Haghighi, Siamak Yassemi, and Rahim Zaare-Nahandi. A generalization of$k$-Cohen–Macaulay simplicial complexes. 2010. Vol. 50. In Arkiv for Matematik. pp.279-290. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203632597718021.
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