Let be a weakly Koszul module. Then Martínez-Villa and Zacharia proved that admitted a filtration of submodules , such that all are Koszul modules (see [11]). Now further, let and be the corresponding minimal graded projective resolutions. We prove that, for all , . Moreover, we also give a new characterization for a module to be weakly Koszul in terms of the filtration of the complex , where is a minimal graded projective resolution.