Derived category of homotopy category of R-module I was wondering if we construct the homotopy category of R-module is it the same as homotopy category of projective R-mod ?
In the homotopy category of projective R-mod we know that quasi-isomorphism are the same as isomorphism, and the derived category of the homotopy category is obtained by formally inverting all the quasi-isomorphism, such that they become isomorphism.
I was wondering what condition is necessarily such that the derived category of R-mod the same as the derived category of projective R-mod ?
 A: In the unbounded situation, the statement I know is the following:
Proposition [Bökstedt–Neeman, Prop. 2.12*]. Let $\mathscr{A}$ be an abelian category with enough projectives satisfying AB4. Then, the composite functor
$$K(P) \hookrightarrow K(\mathscr{A}) \to D(\mathscr{A})$$
is an equivalence of categories, where $K(P)$ is the smallest full subcategory containing the bounded above complexes of projectives that is closed under coproducts and the formation of triangles in the homotopy category $K(\mathscr{A})$.
Bökstedt and Neeman really prove the version involving injectives, but you can dualize appropriately to get the result above. At some point I wrote up some of the relevant details in §1 of this note.
It is also probably important to remark that this category $K(P)$ consists of what are called K-projective complexes in the literature, which were first defined by Spaltenstein, to whom the proposition above could also be attributed (although the result is not stated explicitly). Note that $K(P)$ is not quite the same as the subcategory of $K(\mathscr{A})$ consisting of all complexes with projective terms.
