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Section 2.3 of Hovey's Model Categories book defines a model category structure on Ch(R-Mod), the category of chain complexes of R-modules, where R is a ring. Lemma 2.3.6 then essentially states (I think) that taking projective resolutions of a module corresponds to taking cofibrant replacements of the module, at least in nice cases (e.g. when the projective resolution is bounded below). There is of course also a "dual" model category structure which gives the "dual" result for injective resolutions and fibrant replacements (Theorem 2.3.13).

  1. I think the results in Hovey are proven for not-necessarily-commutative rings. Do things become nicer if we restrict our attention to commutative rings only?

  2. Do these results generalize? For example, is there an analogous model category structure and an analogous result for Ch(OX-Mod), the category of chain complexes of OX-modules, where X is a scheme? More generally, how about for Ch(A), where A is an abelian category?

If the answers to these questions are known, then I assume they would be "standard", but I don't know a reference.

I've re-asked my question in a different form here.

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I assume so as well, though I likewise can't provide a reference. –  Ben Webster Oct 6 '09 at 18:25
The link to the reask of the question is broken (... well, it redirects to a question about numerical methods for calculating digits of pi ...). –  cdouglas Oct 27 '11 at 11:53

2 Answers 2

up vote 8 down vote accepted

I don't think the existence of the dual "injective" model structure merits an "of course," since its generators are much less obvious to construct. However, it turns out that injective model structures actually exist in more generality than projective ones, for instance they exist for most categories of sheaves. I believe this was originally proven by Joyal, but it was put in an abstract context by Hovey and Gillespie.

The basic idea is that model structures on Ch(A) correspond to well-behaved "cotorsion pairs" on A itself. The projective model structure comes from the (projective objects, all objects) cotorsion pair (which is well-behaved for R-modules, but not for sheaves), and the injective one comes from (all objects, injective objects). There is also e.g. a flat model structure coming from (flat objects, cotorsion objects) which is monoidal and thus useful for deriving tensor products. A good introduction, which I believe has references to most of the literature, is Hovey's paper Cotorsion pairs and model categories.

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Awesome, thanks for the reference! –  Kevin H. Lin Oct 13 '09 at 21:30
I would also add the following references: J.D. Christensen and M. Hovey, Quillen model structures for relative homological algebra, Math. Proc. Camb. Phil. Soc. 133 (2002), no. 2, 231-293. The following paper should answer your questions 1) and 2) quite precisely. intlpress.com/HHA/v11/n1/a11 –  Denis-Charles Cisinski Oct 25 '09 at 21:36
The paper of Hovey that I linked to includes a reference to the Christensen-Hovey paper, and to a number of others; I didn't want to give a huge list of references here. Thanks for the other link. –  Mike Shulman Oct 27 '09 at 0:32

I don't know the "standard" answer, but the exact same construction should work for any abelian category with a small projective generator, where "small" means that any map into a sufficiently large well-ordered colimit factors through some stage. This is exactly what is needed to make the small object argument work. Just replace the ring with your small projective generator in the "sphere" and "disk" objects.

For the dual (injective) model structure, cosmall injectives don't tend to exist in practice (for example, they don't exist in abelian groups), so you have to use a more complicated set of maps that I don't understand well and don't know how to generalize. In particular, in the case of quasicoherent sheaves one would need to generalize the injective model structure, and I don't know anything about that.

I don't know of any reason to expect commutative rings to give nicer results.

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This answer also fits into the framework of Gillespie's work. Asking for a small projective generator is related to asking for the cotorsion pair $(A,B)$ of Mike's answer to be cogenerated by a one-element set (cogenerated by a set means there is a set $S\subset A$ such that $b\in B$ iff $Ext^1(S,b)=0$). In a Grothendieck category with enough projectives, being cogenerated by a set forces $(A,B)$ to be a complete cotorsion pair. You'll also get the dual complete cotorsion pair, and thence the Hovey model structure described by Mike. –  David White Aug 8 '12 at 21:41

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