Is a chain complex with fin-gen homology generally (quasi-)isomorphic to one with fin-gen modules in each index? Let $R$ be a left Noetherian ring. (By an $R$-module, by default we mean a left one.) We will take chain complexes mentioned below to be of $R$-modules. We denote by $\mathrm{Ch}R$ the category of chain complexes of $R$-modules.
Let $C$ denote the full subcategory of $\mathrm{Ch}R$ consisting of those chain complexes which have all their homology modules finitely-generated over $R$.
Let $D$ denote the full subcategory of $\mathrm{Ch}R$ consisting of those chain complexes which have a finitely-generated $R$-module in every index.
Because we assumed $R$ (left) Noetherian, we have $D \subset C$.

My questions are as follows:

*

*Is every object of $C$ quasi/weakly isomorphic (as a chain complex of $R$-modules) to an object of $D$?


*If yes, then more generally, given a category $J$ and a diagram/functor $F : J\rightarrow C$, is $F$ naturally quasi-isomorphic in $\mathrm{Ch}R$ to some functor $F' : J\rightarrow D$?


*And if yes, do the above statements fail if we require actual isomorphisms (of chain complexes), instead of just quasi-isomorphisms?
 A: The answer in general is no.
This question, and related matters, is addressed in the nice recent paper
Positselski, Leonid; Schnürer, Olaf M., Unbounded derived categories of small and big modules: is the natural functor fully faithful?, J. Pure Appl. Algebra 225, No. 11, Article ID 106722, 23 p. (2021). ZBL1464.18015.
In Theorem 3.1 of this paper, it is proved that if $R$ is a quasi-Frobenius ring of infinite global dimension, then there is a complex $X$ of $R$-modules with all homology modules finitely generated but which is not quasi-isomorphic to a complex of finitely generated modules.
The description of $X$ is not too hard, but the proof that it is not quasi-isomorphic to a complex of finitely generated modules is non-trivial and ingenious.
As an example, take $R=\mathbb{Z}/4\mathbb{Z}$ and take $X$ to be the direct sum of the sequence of complexes
\begin{align}
\dots\rightarrow 0\rightarrow 0\rightarrow 0\rightarrow R\rightarrow R&\rightarrow R\rightarrow 0\rightarrow 0\rightarrow 0\rightarrow\dots\\
\dots\rightarrow 0\rightarrow 0\rightarrow R\rightarrow R\rightarrow R&\rightarrow R\rightarrow R\rightarrow 0\rightarrow 0\rightarrow\dots\\
\dots\rightarrow 0\rightarrow R\rightarrow R\rightarrow R\rightarrow R&\rightarrow R\rightarrow R\rightarrow R\rightarrow 0\rightarrow\dots\\
\vdots&
\end{align}
where all the differentials $R\rightarrow R$ are multiplication by $2$.
The homology of $X$ is finitely generated, since it is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ in every degree except the middle one, where it is zero.
The same paper also has some positive results. For example, in Corollary 4.5 it is shown that the answer to question 1 is yes when $R$ has finite left global dimension.
