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 homotopic 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?

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?