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Hallo,

I have to worry you one more time with these acyclicity problems, but as I am currently working on derived functors in a.g., I really need to understand derived functors in a very general form.

So my question is just: I have an exact functor $F: K^{+}(A)\rightarrow K(B)$ of the homotopy categories of abelian categories A and B and I know that the derived $RF: D^{+}(A)\rightarrow D(B)$ exists (in the sense of: it is exact and has universal property). Then this does not a priori imply that for each complex in $K^{+}(A)$ I can find a quasiiso to a complex of F-acyclics? I myself would guess that one has to have a triangulated subcategory L of $K^{+}(A)$ which is adapted to F, in the sense of Hartshorne, Residues and Duality. A short hint to if I am right is totally enough, thanks.

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    Why do you want to do this?2011-08-07
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    To follow up on Mariano's question: Can you give an example where you find yourself this situation? Moreover, I don't think Hartshorne's R&D uses the word *adapted* at all (I think this appears in Gelfand-Manin only). What is your main reference and what do you actually want to do? Why are the references you were given so far not sufficient?2011-08-08
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    That's right, Hartsorne just speaks of a triangulated subcategory L "as in Theorem I, 5.1.". So as an abbreviation I would like to call such L simply adapted, as in Gelfand-Manin. Well, I want to know this because it seems to me of principal interest: in the simple situation when you have a left exact functor $F:A\rightarrow B$ and an F-adapted class, then you know that the derived exists. Furthermore one knows that then the class of F-acyclic objects also is an F-adapted class, i.e. one can find acyclic resolutions. I am asking whether the existence of the derived implies already enough acycs2011-08-08
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    Okay, you clarified the terminology and the academic motivation. Still, you haven't addressed my main question: can you give me an example in which you know that the derived functor exists and you *don't* know that there exist enough acyclics?2011-08-08
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    Well, I don't know such an example, but that's exactly my question: can there be such examples or does the existence of the derived automatically imply enough acyclics.2011-08-08

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