We say that a group $G$ is of type $FL$ if there exists a resolution $L_\bullet \to \mathbb{Z}$ of finite length of finitely generated, free $\mathbb{Z}G$-modules. Now, un unproven proposition in Brown's book "Cohomology of groups" says that if there exists a finite complex $X$ which is a $K(G,1)$, then $G$ is of type $FL$. I thought immediately to the cellular homology chain $\ldots\to H_{n+1}(X_{n+1},X_n)\to H_n(X_n,X_{n-1})\to\ldots$ where $X_n$ is the $n$-skeleton of $X$, but I am not sure about it and I don't know how to equip these free groups with a $G$-module structure. I suppose that I should use that $G\cong\pi_1(X,x_0)$ but I would like to know, at least, if I'm on the right way or not.
Thank you in advance, bye