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In this paper we mainly consider local rings admitting dualizing complexes. It is well-known that if a Noetherian local ring A admits a dualizing complex, then the non-Cohen–Macaulay (abbreviated CM) locus of A is closed in the Zariski topology (cf. [8, 10]). If the dimension of this locus is zero and A is equidimensional, i.e. the punctured spectrum of A is locally CM and dim(A/P) = dim (A) for all minimal prime ideals P ∈ Ass (A), then A is a generalized CM ring and its structure is well-understood (see [2, 12]). For instance, one of the characterizations of generalized CM rings is the conditions that for any parameter ideal q contained in a large power of the maximal ideal m of A, the difference between length and multiplicityis independent of the choice of q. However, if the dimension of the non-CM locus is larger than zero, little is known about how this dimension is related to the structure of the local ring A. The purpose of …