The author obtains a generalization of M. Hochster’s formula [in: Ring Theory II, Proc. 2nd Okla. Conf. 1975, 171–223 (1977; Zbl 0351.13009)] for local cohomologies of square-free monomial ideals to monomial ideals that are not necessarily square-free and analyzes conditions under which local cohomology groups vanish. In particular, the vanishing degrees of generalized CM monomial ideals are determined. This allows him to deduce combinatorial characterizations of generalized CM monomial ideals in terms of the exponents of variables in the monomial generators. Making use of the generalized Hochster’s formula, the author compares local cohomologies for such an ideal and its radical.
Based on the combinatorial characterization of generalized CM property, he then describes a method of construction of generalized CM monomial ideals from Buchsbaum Stanley-Reisner ideals and considers some examples.