Equality of maps in the derived category I have two chain complexes $C$ and $D$ of $R$-modules over some ring $R$ and two chain maps $f,g: C \to D$. I know that both $f$ and $g$ induce the same map in homology, i.e. $f_* = g_*$. When can I say that $f$ and $g$ are equal in the derived category $\text{D($R$-Mod)}$?
I think this is always true when the global dimension of $R$ is $<=1$, since you can build a quasi-isomorphism $C \to H_*(C)$. I'm specifically interested in the case where $R$ is a multivariable polynomial ring over $\mathbb{Q}$, however, where I believe it is not always true. Is there a nice way to check this for a specific choice of $f,g$?
Also, I'd be interested to see an example of a case where this isn't true, just because I can't seem to find one or come up with one myself.
Any advice would be appreciated! Thank you.

Even if $R$ has global dimension $1$ this is not always true.
For example, let $R=\mathbb{Z}$, let
$$C:= \cdots\xrightarrow{}0\xrightarrow{}\mathbb{Z}\xrightarrow{\times2}\mathbb{Z}\xrightarrow{}0\xrightarrow{}\cdots,$$
and let
$$D:= \cdots\xrightarrow{}0\xrightarrow{}\mathbb{Z}/2\mathbb{Z}\xrightarrow{}0\xrightarrow{}0\xrightarrow{}\cdots.$$
Then there are two chain maps $C\to D$, one of them zero and the other nonzero in the derived category, but both induce the zero map on homology.
