# 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?

-
When you speak about chain homotopic complexes, do you mean homotopy equivalent instead? Homotopy equivalent complexes have the same (co)homology with any coefficients, so if one of the is a projective resolution, the other one, even if it weren't, would do the same job. – Fernando Muro Apr 29 '13 at 21:27
I mean chain homtopy equivalent. I realize they do the same job from the point of view of computing derived functors but I would still like to know if the complex is projective. – Benjamin Steinberg Apr 29 '13 at 22:54
If $U$ is any non projective module, you can add to a projective resolution of $M$ a complex of the form $\cdots\to0\to U\to U\to0\to\cdots$ (with the non-trivial map the identity) and this will be homotopy equivalent to the original resolution. – Mariano Suárez-Alvarez Apr 29 '13 at 23:33
Can you say any more if the chain homotopy from the projective resolution to the other resolution is surjective at each chain module? – Benjamin Steinberg Apr 30 '13 at 17:25

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.