Existence of a bounded finitely generated torsion-free resolution

Question

I am looking for a reference for (or a proof of) the following fact:

Let $G$ be a profinite group. Let $X^\bullet$ be a complex of discrete $G$-modules. We assume that the cohomology $G$-modules of $X^\bullet$ are nontrivial only for finitely many degrees, and that they are finitely generated over $\mathbb{Z}$. (We do not assume that the $G$-modules $X^i$ are finitely generated over $\mathbb{Z}$.) Then there exists a quasi-isomorphism $M^\bullet \to X^\bullet$, where $M^\bullet$ is a bounded complex of finitely generated (over $\mathbb{Z}$) torsion-free $G$-modules.

Answer 1

Proof (due to Joseph Bernstein). Assume that $H^i(X^\bullet)=0$ for $i>n$. We choose a $G$-morphism $A^n\to \ker[X^n\to X^{n+1}]$ such that the induced morphism $A^n\to H^n(X^\bullet)$ is surjective, where $A^n$ is a finitely generated (over $\mathbb{Z}$) torsion-free $G$-module. We regard $A^n$ as a complex (with one $G$-module $A^n$ in degree $n$). We have a morphism of complexes $\varphi\colon A^n\to X^\bullet$. We denote by $X_{(1)}^\bullet$ the cone of $\varphi$. It is easy to see that $H^n(X_{(1)}^\bullet)=0$. Then we apply this procedure to $X_{(1)}^\bullet$ for $n-1$ to obtain $X_{(2)}^\bullet$ with $H^{n-1}(X_{(2)}^\bullet)=0$, and so on.

Assume that $H^i(X^\bullet)=0$ for $i\le n-m$. Then the complex $X_{(m)}^\bullet$ is acyclic. One can check that $X_{(m)}^\bullet$ is the cone of some morphism of complexes $\psi\colon M^\bullet\to X^\bullet$, where $M^\bullet$ is a bounded complex of finitely generated torsion-free $G$-modules. Since the cone $X_{(m)}^\bullet$ of $\psi$ is acyclic, we see that $\psi$ is a quasi-isomorphism.