A generalization of k-Cohen–Macaulay simplicial complexes

Abstract

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.