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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?

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  • $\begingroup$ 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. $\endgroup$
    – Fernando Muro
    Commented Apr 29, 2013 at 21:27
  • $\begingroup$ 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. $\endgroup$
    – Benjamin Steinberg
    Commented Apr 29, 2013 at 22:54
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    $\begingroup$ 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. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Apr 29, 2013 at 23:33
  • $\begingroup$ Can you say any more if the chain homotopy from the projective resolution to the other resolution is surjective at each chain module? $\endgroup$
    – Benjamin Steinberg
    Commented Apr 30, 2013 at 17:25

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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.

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