Let $V$ and $V^\prime$ - complexes of modules over ring $A$, and $f, g$ - homomorphisms $V\rightarrow V^\prime$.

I am interested in various conditions on $A, V, V^\prime$: ($f$ and $g$ are homological) $\Rightarrow$ ($f$ and $g$ are homotopic).

(I knew one example: $A$ - hereditary algebra and $V, V^\prime$ - complexes of projective modules, bounded from the right. But recently I understood that in this case it's not true that ($f$ and $g$ are homological) $\Rightarrow$ ($f$ and $g$ are homotopic))