Existence of a bounded finitely generated torsion-free resolution 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.
 A: 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.
