Does localization at quasi-isomorphisms imply homotopy invariance? Usually, the derived category of some abelian category $A$ (I'm happy already with $A$-mod) is defined first taking chain complexes up to homotopy, and then localize at quasi-isomorphisms.
My question is, if one begins with $Chain(A)$=complexes in $A$ (instead of complexes up to homotopy equivalence) , and then localize at quasi-isomorphisms, do we get the same?
Denote
$$Chain(A)\overset{\tilde Q}{\to} [qis]^{-1}Chain(A)$$
the functor with the universal property w.r.t. this localization.
It is obvious that an homotopy equivalence is a quasi-isomorphism, but is it also obvious (or is it true?) that if $f\sim g$ are two homotopic maps then $Q(f)=Q(g)$? Is this category the same as the usual derived category?
 A: Turning a comment into an answer: a pair of homotopic maps $A \longrightarrow B$ is the same as a single map from a cylinder of the identity map of $A$ into $B$. This cylinder is then homotopy equivalent to $A$ in two ways, and using this one can show that any functor that sends homotopy equivalences into isomorphisms (such as localization functor $\tilde Q$) factors uniquely through the homotopy category. In other words, functor from the category of complexes into the homotopy category is a localization. It follows that quasi-isomorphism localizations of the category of complexes and of the homotopy category are equivalent.
The advantege of going through a two-step process is that the homotopy category is  already triangulated, and the Verdier localization procedure for triangulated categories is somewhat simpler than the general Gabriel-Zisman. For example, it's not obivious that the derived category will be triangulated, if you won't go through this intermediate step.
The sources for that are, for example, chapter 10 of Weibel or chapter 3 of Gelfand-Manin.
