Let $\cal{A}$ be an abelian category with enough projectives and $\mathbf{C}_+ (\cal{A})$ the category of bounded below chain complexes.

Since Quillen (Homotopical algebra, 1.2, examples B), there is a well-known "standard" model category structure on $\mathbf{C}_+ (\cal{A})$ taking as weak equivalences the maps inducing isomorphisms on homology, as fibrations the degree-wise epimorphisms in $\cal{A}$ and with cofibrations maps $i$ which are injective and such that $\mathrm{cok}\ i$ is a complex having a projective object of $\cal{A}$ in each degree.

More recently, Hovey (Model categories), proved an analogous result for the category of *unbounded* chain complexes, but with ${\cal A} = R$-modules, $R$ a ring (but cofibrations are not so easy to characterise). Finally, it's folklore (at least, I don't know if it is published somewhere) that the same holds for $\cal{A}$ an abelian category with a projective generator -the fact that allows the small object argument to work, as Eric Wofsey points out in his answer to this MO question.

I'm interested in the following variant of this problem: is there a model structure on $\mathbf{C}_+ (R)$ taking as weak equivalences the

homotopyequivalences?

If it's true, I think this should be easy to verify: just taking a look to the classical proof and seing if you can change "**homology** equivalences" everywhere by "**homotopy** equivalences". I'm willingly going to do it, but, prior to start, I would like to know if it is already done, much as in the case of topological spaces where, together with the "standard" (Quillen too) model structure with **weak homotopy** equivalences as weak equivalences, there is the Strom model structure (The homotopy category is a homotopy category), with **homotopy** equivalences as weak equivalences.