I'm studying homotopical algebra and I'm trying to prove the following fact:

If $R$ is a quasi-Frobenius ring (for $R$-modules one has "projective module$\iff$ injective module") then the stable category of $R$-modules is (equivalent to) the homotopy category of $R$-mod, once the latter has been equipped with the model structure in which cofibrations are monomorphisms, fibrations are epimorphisms and weak equivalences are stable-isomorphisms. In order to do this, since in this model structure each module is both fibrant and cofibrant, it's enough to show that the homotopy relation is the same as the stable-equivalence relation. More precisely one needs to show that $f$ is left (or right, it's the same) homotopic to $g$ $\iff$ $f$ is stable equivalent to $g$.

Now my problem is in the "$\Leftarrow$" part: how can I construct a left, say, homotopy between $f$ and $g$ once I know that $f-g$ factorizes through a projective (hence also injective) $R$-module? Thanks in advance.


I had this idea:

Denote with $\pi:Mod_R\twoheadrightarrow Ho(Mod_R)$ the projection-localization functor to the homotopy category. Since $0\rightarrowtail M\twoheadrightarrow 0$ every module is both fibrant and cofibrant (or bifibrant), and thus $Ho(Mod_R)$ is isomorphic (not just equivalent) to the classical homotopy category of $Mod_R$ (I guess you read Hovey's book "Model Categories", as it's the only reference I know on this subject, so you're probably familiar with this terminology). In particular two maps $f,g:M\rightarrow N$ are (left-right) homotopic if and only if they are identified by $\pi$.

Moreover, notice that a module $B$ is projective-injective (or bijective) if and only if $0\rightarrow B$ is a trivial cofibration, and if and only if $B\rightarrow 0$ is a trivial fibration: in particular it's a zero object in the Homotopy category.

Then if the morphism $f-g:M\rightarrow N$ factors through a bijective object, necessarily $\pi(f-g)=0$, so all you have to show is that $\pi$ is an additive functor (you can find the proof here).


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