In Part 1, we describe six projective-type model structures on the categoryof differential graded modules over a differential graded algebra A over acommutative ring R. When R is a field, the six collapse to three and arewell-known, at least to folklore, but in the general case the new relative andmixed model structures offer interesting alternatives to the model structuresin common use. The construction of some of these model structures requires twonew variants of the small object argument, an enriched and an algebraic one,and we describe these more generally. In Part 2, we present a variety oftheoretical and calculational cofibrant approximations in these modelcategories. The classical bar construction gives cofibrant approximations inthe relative model structure, but generally not in the usual one. In the usualmodel structure, there are two quite different ways to lift cofibrantapproximations from the level of homology modules over homology algebras, wherethey are classical projective resolutions, to the level of DG-modules overDG-algebras. The new theory makes model theoretic sense of earlier explicitcalculations based on one of these constructions. A novel phenomenon weencounter is isomorphic cofibrant approximations with different combinatorialstructure such that things proven in one avatar are not readily proven in theother.
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