Let $A$ be a ring and $f \in A$ an element. If $M$ is an $A$-module on which multiplication by $f$ is an isomorphism, then $M$ is in fact an $A\_f$-module. 


Now suppose that $C \in D(A)$ is a complex in the derived category of $A$-modules and that the multiplication by $f$ map $C \xrightarrow{f} C$ is a (quasi-)isomorphism. Does $C$ necessarily come from $D(A\_f)$? That is, is there some complex $C'$ of $A\_f$-modules and a (quasi-)isomorphism $C' \rightarrow C$ of complexes of $A$-modules. What if $C$ is bounded or bounded above?



Characterizing the image of $D(A_f) \rightarrow D(A)$
The two sides of your "that is" are different questions:


1) Does $C$ necessarily come from $D(A\_f)$ ?


2) Is there some complex $C'$ of $A\_f$ modules and a quasi-isomorphism $C'\rightarrow C$ of complexes of $A$-modules?


Regarding 1): By assumption, $f$ induces an isomorphism on homology. Therefore, by your first observation, the homology of $C$ is already an $A\_f$ module. Therefore this homology won't change if we localize at $f$. Therefore the map $C\rightarrow C\_f$ is a quasi-isomorphism, which shows that in the derived category, $C$ is quasi-isomorphic to an object that comes from $D(A\_f$).


This, however, constructs a map $C\rightarrow C'$, not a map $C'\rightarrow C$, and therefore does not speak to question 2).


