Weakly Koszul-Like Modules

The Koszul-like property for any finitely generated graded modules over a Koszul-like algebra is investigated and the notion of weakly Koszul-like module is introduced. We show that a finitely generated graded module $M$ is a weakly Koszul-like module if and only if it can be approximated by Koszul-like graded submodules, which is equivalent to the fact that $\mathbf G(M)$ is a Koszul-like module, where $\mathbf G(M)$ denotes the associated graded module of $M$. As applications, the relationships between the minimal graded projective resolutions of $M$ and $\mathbf G(M)$, and the Koszul-like submodules are established. Moreover, the Koszul dual of a weakly Koszul-like module is proved to be generated in degree $0$ as a graded $E(A)$-module.