Cofibrant Replacement of chain complexes

Question

Hi,

I encountered an issue today that I can't resolve to myself:

Consider the projective model structure on chain complexes over a ring R (Ch(R)), bounded below if you like.

Projectives in Ch(R) are split-exact complexes which are level-wise projective (so in particular are contractible). Cofibrations in Ch(R) are inclusions with projective cokernel, and so every cofibrant object is projective (since the initial object is the zero complex). In particular every cofibrant object is contractible, and so since every object of Ch(R) has a cofibrant replacement, every chain complex has trivial homology????

This obviously can't be right, and I'm misunderstanding something, and any help to figure out what that is would be much appreciated.

Thanks,

Tom

Addition: After thinking about it for a little bit longer I think that the word "projective" in the definition of a cofibration in the projective model structure must mean "level-wise projective", as this resolves the question, and there is a lot of literature on constructing "almost projective" resolutions in derived categories, of which a level-wise projective resolution is an example. I think the reason for their use is that to ask for a (categorically) projective resolution in derived categories is usually too strong a requirement to be able to do anything with it, (as illustrated by the contractibility of projective objects in Ch(R)). I hope this paragraph is correct, but comments/answers/corrections still very welcome.

Answer 1

The "addition" to your own question seems to be a starting point for the answer you were looking for but let me add some comment. In particular, the projective model structure you mention is a very particular model structure which has the good fortune to be an abelian model structure in the sense of [M. Hovey, Cotorsion pairs and model categories, Contemporary Maths. (2006)]. In particular, in the projective model structure on chain complexes, the cofibrations are the monomorphisms with cofibrant (=DG-projective) cokernel, the fibrations are the epimorphisms, and the weak equivalences are the quasi-isomorphisms. So the notion of "projective" complex you are looking for is that of DG-projective complex, that is, a complex in which each entry is projective, and such that any map from it to an exact complex is chain homotopic to 0 (in the bounded case this is equivalent to level-wise projective so your intuition is correct).

I suggest you to read the very well written paper by Hovey that I mentioned above and to use the papers in his reference list for further details. Of course, Hovey's book on model categories is a standard reference but the paper I'm suggesting is far easier and faster to read.