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Let $\newcommand{\A}{\mathscr{A}}\A$ be a grothendieck abelian category. In their paper "Local and stable homological algebra in grothendieck abelian categories", Cisinski and D├ęglise define the notion of a descent structure on $\A$, to which corresponds a model structure on the category $C(\A)$ of complexes, where weak equivalences are quasi-isomorphisms. See the nlab page for some background.

The example I am interested in is the case where $\A$ is the category of modules over a ringed space. In example 2.3, they give a descent structure on this category which they show is flat (example 3.1), which means that it induces a symmetric monoidal model structure on $C(\A)$ (proposition 3.2). This is interesting because you automatically get a derived tensor product on the homotopy category (i.e. the derived category of $\A$); the usual injective model structure is not monoidal.

At the end of the day I am interested in the derived category $D(\A)$, so I was wondering about the derived functors you get this way ($Lf^*$, $Rf_*$, $\otimes^L$ etc). Are they basically the same as the usual ones, as defined for example in Lipman's book? That is, are cofibrant and fibrant replacements really the same as taking resolutions? Is it possible to get an explicit description for the cofibrant and fibrant objects in this model category?

I'm a beginner at model categories so I hope this question is not too elementary for this site.

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The explicit construction of derived functors is usually done ny taking fibrant or cofibrant resolutions and then applying the functors you want to derive. Nevertheless, derived functors are characterized by universal properties which only refer to the homotopy category and to the projection functor (see Grothendieck's notion of total derived functor), hence they only depend on what weak equivalences you take, so in your case the answer is yes. – Fernando Muro Jul 6 '13 at 8:54
Prof. Muro, hm yes I had forgotten that the derived functors depend only on weak equivalences. Thanks! I'm still curious about an explicit description of cofibrant / fibrant objects in this specific model structure though, if one can be given. – Adeel Khan Jul 6 '13 at 12:07
Several people have studied the different model structures on categories of chain complexes. Fibrant and cofibrant objects usually correspond to different kinds of resolutions, eg protective, flat, etc. They are connected to torsion theories. I think that Hovey started this. – Fernando Muro Jul 6 '13 at 23:20
Yes, I've looked briefly at Hovey and Gillespie's approaches using cotorsion pairs. Actually my suspicion is that the monoidal model structure you get from the approach of Cisinski and Déglise is the same one you get with cotorsion pairs (e.g. the end of However, I am not sure how to prove this. – Adeel Khan Jul 7 '13 at 8:06
I'm writing from a very poor device now, I don't have access, but there's a lot of literature on the topic, maybe there's even an explicit comparison by now. – Fernando Muro Jul 7 '13 at 8:21

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