2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73420The Hilbert series of local cohomologies for monomial ideals, which are not necessarily square-free, is established. As applications, we give a sharp lower bound of the non-vanishing degree of local cohomologies and also a sharp lower bound of the positive integer k of k-Buchsbaumness for generalized Cohen-Macaulay monomial ideals.Commutative Algebra13D45, 13F20, 13F55text