Resolutions chain homotopic to projective ones Motivation. In my research I have a situation where a monoid $M$ is acting by nice cellular maps on a contractible cell complex and so the augmented chain complex is a resolution of the trivial module over $\mathbb ZM$. Now I can show by a somewhat messy argument that this resolution is projective. I have a much smoother proof that the augmented chain complex of the barycentric subdivision is a projective resolution. These two chain complexes are chain homotopy equivalent as chain complexes of $\mathbb ZM$-modules by the naturality of the chain homtopy equivalence between the augmented cellular chain complexes of a regular cell complex and its barycentric subdivision. In a hope to avoid the messy computation I naively ask the following question. 


Question. If $R$ is a unital ring and $M$ is an $R$-module, what can be said about a resolution of $M$ which is chain homotopy equivalent to a projective resolution of $M$?  Is there any chance it to must be projective?


 A: Here is a conceptual answer that tells you that although resolutions
chain homotopy equivalent to projective resolutions need not 
be projective, they are perfectly acceptible to homological algbra.
As with any form of algebraic homotopy theory, it is very nice
to develop homological algebra in the context of model categories.
In the case of chain complexes of $R$-modules, the identity functor
specifies a Quillen equivalence between two model structures with
the same weak equivalences, namely the quasi-isomorphisms.  In the one
in most common use (called the $q$-model structure in ``More concise
algebraic topology'', by Kate Ponto and myself), a $q$-cofibrant
approximation of an $R$-module $M$ (viewed as a chain complex 
concentrated in degree zero) is a projective resolution of $M$. In the other (called the $m$-model structure, opus cit), the 
$m$-cofibrant objects are those of the chain homotopy type of 
$q$-cofibrant objects, so an $m$-cofibrant approximation is the
up-to-homotopy version of a projective resolution you are asking
about.  The Quillen equivalence is a kind of high level way of 
saying that the two kinds of resolutions can be used interchangably.
