Let $M$ be a weakly Koszul module. Then Martínez-Villa and Zacharia proved that $M$ admitted a filtration of submodules $0 = U_0 \subseteq U_1 \subseteq U_2 \subseteq \dots · · · \subseteq U_p = M$, such that all $U_{i+1}/U_i$ are Koszul modules (see [11]). Now further, let $\mathcal P_∗^i \to U_i/U_{i−1} \to 0$ and $\mathcal P_∗ \to M \to 0$ be the corresponding minimal graded projective resolutions. We prove that, for all $n \geqslant 0$, $\mathcal P_n \cong \oplus \mathcal P_n^i$ . Moreover, we also give a new characterization for a module $M$ to be weakly Koszul in terms of the filtration of the complex $\mathcal P_∗$, where $\mathcal P_∗ \to M \to 0$ is a minimal graded projective resolution.