Let $\mathcal{A}$ be an abelian category, $D(\mathcal{A})$ be its derived category and $X,Y$ be complexes with morphisms in $\mathcal{A}$. I am trying to understand what does it mean to say that $X$ and $Y$ are isomorphic in $D(\mathcal{A})$. Obviously, if there is a quasi-isomorphism $X \longrightarrow Y$, then these complexes are isomorphic in $D(\mathcal{A})$, but does the converse hold? I have two questions:
- If $X$ and $Y$ are isomorphic in $D(\mathcal{A})$, then is it true that there is a quasi-isomorphism $X \longrightarrow Y$?
- If $X$ and $Y$ are isomorphic in $D(\mathcal{A})$, then there is a (left) fraction $(s,f) : X \longrightarrow Y$, where $s : Z \longrightarrow X$ is a quasi-isomorphism and $f: Z \longrightarrow Y$ is a morphism of complexes (modulo homotopy), such that $(s,f)$ is an isomorphism in $D(\mathcal{A})$. In this case, is it true that $f$ is also a quasi-isomorphism?
I will be also grateful for any explanation or idea which may clarify the meaning of two complexes being isomorphic in the derived category.