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?

finitely generatedmodules. Is that what you intended? $\endgroup$ – Steven Landsburg Jun 21 '16 at 1:24