Cfrgtkky

Let $C_*$, $D_*$ be chain complexes of modules over a ring $R$. Suppose that $fcolon C_* rightarrow D_*$ is a quasi-isomorphism (i.e. an isomorphism in Homology).

I am wondering what conditions are needed (over either $R, C_*$ or $D_*$) so that the dual chain complexes $hom_R(C_*,R)$ and $hom_R(D_*, R)$ are quasi-isomorphic.

$newcommand{Hom}{mathrm{Hom}} require{AMScd}$
Suppose $C$ and $D$ are chain complexes of free $R$-modules, where $R$ is a PID. If $f: C to D$ is a homology isomorphism, then $f^*: Hom(D,R) to Hom(C,R)$ is as well. This follows from the naturality of the Universal Coefficient Theorem, as there is a commuting diagram of exact sequences
$$ begin{CD} 0 @>>> Ext(H_{ast-1}(D),R) @>>> H^ast(Hom(D,R)) @>>> Hom(H_ast(D),R) @>>>0 \
@VVV @VVV @VVV @VVV @VVV\
0 @> >>Ext(H_{ast-1}(C),R)@>>> H^ast(Hom(C,R)) @>>>Hom(H_ast(C),R) @>>>0end{CD}$$
where the vertical maps are all induced by $f: C to D$. As $f$ is a homology isomorphism, all maps except the middle are isomorphisms. Hence, the Five Lemma gives that the middle map is an isomorphism as well.

For more general $R$, there should be a spectral sequence instead of this short exact sequence giving the same result.