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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(O_X-Mod), the category of chain complexes of O_X-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.

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(O_X-Mod), the category of chain complexes of O_X-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.

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(O_X-Mod), the category of chain complexes of O_X-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.

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(O_X-Mod), the category of chain complexes of O_X-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.