I am trying to refine my understanding of derived categories.
Let $\text{Mod}_R$ and $\text{Mod}^f_R$ be respectively the categories of modules and finitely generated modules over a Notherian ring $R$ (commutative with unit). I would like to show that, given $N$ and $M$ two finitely generated $R$-modules, the extension modules $\operatorname{Ext}^i(N,M)$ in both $\text{Mod}_R$ and $\text{Mod}^f_R$ coincide. One reason for this statement would be that $\text{Mod}_R$ do have enough injective, whereas $\text{Mod}^f_R$ does not in general. My question is a bit more precise:
Let $\operatorname{C}^{\cdot}$ and $\operatorname{C}_f^{\cdot}$ (resp. $\operatorname{K}^{\cdot}$ and $\operatorname{K}_f^{\cdot}$ or $\operatorname{D}^{\cdot}$ and $\operatorname{D}_f^{\cdot}$) be the categories of chain complexes (resp. homotopy or derived categories) of $\text{Mod}_R$ and $\text{Mod}^f_R$ respectively, where $\cdot$ refers to the boundedness condition (that is $\cdot=+,-,b$ or $\emptyset$). The embedding $\operatorname{C}^{\cdot}_f \hookrightarrow \operatorname{C}^\cdot$ - induced by $\text{Mod}_R^f\hookrightarrow \text{Mod}_R$ - commutes the cohomological functors $H^i$, and hence defines a full and faithful embedding $\operatorname{K}^{\cdot}_f \hookrightarrow \operatorname{K}^\cdot$ of the homotopy categories. Because it is exact, it induces a functor $\operatorname{D}^{\cdot}_f \rightarrow \operatorname{D}^\cdot$. As such, given a bounded complex $N^\bullet$ in $\operatorname{C}^b_f$ and a complex $M^{\bullet}$ in $\operatorname{C}^{\cdot}_f$, we have a natural morphism of $R$-modules: $$ \operatorname{Hom}_{\operatorname{D}^\cdot_f}(N^\bullet,M^\bullet)\longrightarrow \operatorname{Hom}_{\operatorname{D}^\cdot}(N^\bullet,M^{\bullet}).$$ Then, is the above an isomorphism? The original question would be recovered by taking $N^\bullet=N[0]$ and $M^\bullet=M[i]$.
Unwinding definitions, it seems that I am asking for the following: for any quasi-isomorphism $L^\bullet \to K^\bullet$ in $D^\cdot$ where $L$ is a complex of finitely generated $R$-modules, there exists a complex of finitely generated $R$-modules $H^\bullet$ and a quasi-isomorphism $K^\bullet\to H^\bullet$. This seems quite strong to me, right?
