A simplicial complex of dimension d - 1 is said to be Cohen-Macaulay in codimension t, 0 <= t <=d -1, if it is pure and the link of any face with cardinality at least t is Cohen-Macaulay. This generalizes several concepts on simplicial complexes such as Cohen-Macaualyness, Buchsbaum property, S_r condition of Serre, and locally Cohen-Macaulayness. Most results on the simplicial complexes with aforementioned properties naturally extend to the case of Cohen-Macaulayness in codimension t. In particular, the Eagon-Reiner theorem, the local behavior, and the homological vanishing properties are suitably retained. Furthermore, characterizations of certain families of Cohen-Macaulay simplicial complexes carry over characterizations of these families of simplicial complexes which are Cohen-Macaulay in codimension t. This talk is based on recent joint works with H. Haghighi, S. A. S. Fakhari and S. Yassemi. 1

A simplicial complex of dimension d - 1 is said to be Cohen-Macaulay in codimension t, 0 <= t <=d -1, if it is pure and the link of any face with cardinality at least t is Cohen-Macaulay. This generalizes several concepts on simplicial complexes such as Cohen-Macaualyness, Buchsbaum property, S_r condition of Serre, and locally Cohen-Macaulayness. Most results on the simplicial complexes with aforementioned properties naturally extend to the case of Cohen-Macaulayness in codimension t. In particular, the Eagon-Reiner theorem, the local behavior, and the homological vanishing properties are suitably retained. Furthermore, characterizations of certain families of Cohen-Macaulay simplicial complexes carry over characterizations of these families of simplicial complexes which are Cohen-Macaulay in codimension t. This talk is based on recent joint works with H. Haghighi, S. A. S. Fakhari and S. Yassemi. 1