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I had a few questions about compositions of derived functors, the comparison between hypercohomology, and sheaf cohomology and the following theorem from the Gelfand, Manin homological algebra book:

Theorem (Methods of homological algebra. Gelfand, Manin. Page 200). Let $\mathcal{A}, \mathcal{B}, \mathcal{C}$ be three abelian categories, $F: \mathcal{A} \rightarrow \mathcal{B}$, $G: \mathcal{B} \rightarrow \mathcal{C}$ be two additive left exact functors. Let $\mathcal{R_\mathcal{A}} \subset Ob \ \mathcal{A}$ (resp. $\mathcal{R}_\mathcal{B} \subset Ob \ \mathcal{B}$) be a class of objects adapted to $F$ (resp. to $G$). Assume that $F(\mathcal{R}_\mathcal{A}) \subset \mathcal{R}_\mathcal{B}$. Then the derived functors $RF$, $RG$, $R(G \circ F): \mathcal{D}^+(\bullet) \rightarrow \mathcal{D}^+(\bullet)$ exist and the natural morphism of functors $R(G \circ F) \rightarrow RG \circ RF$ is an isomorphism.

What I wanted to know was:

1 - I basically want to know if the isomorphism $R(G \circ F) \rightarrow RF \circ RF$ applies vertically term by term, that is if it is $R^n(G \circ F) \cong R^nF \circ R^nF$, or if it is $R^{p+q}(G \circ F) \cong R^pF \circ R^qF$? I don't know if this computation is straightforward like this or WHEN do I have to use a spectral sequence?

The second is I'm trying to understand hypercohomology in the context of derived functors and in comparison to sheaf cohomology in order to apply this theorem to hypercohomology: You know how for derived functors you have an object $A$ in an abelian category $A \subset Ob \ \mathcal{A}$ (say a sheaf in the category of sheaves on a topological space), and you compute the derived functor of a left exact functor (the global section functor in this case) right? So my questions were:

2 - Is the hypercohomology functor a derived functor?

3 - Is the right derived construction for sheaf cohomology functors basically interchangeable when it comes to hypercohomology? What I mean is that, in hypercohomology am I basically computing the derived functor of a global section functor $F$ just like in the sheaf cohomology case? The functor being $F: Kom^+(\mathcal{A}) \rightarrow \mathcal{B}$ (for some abelican category $\mathcal{A}$) ? Where the object I'm computing the hypercohomology for is a chain complex $K^\bullet \subset Ob \ Kom^+(\mathcal{A})$, and instead of the typical injective resolutions I have the Cartan-Eilenberg resolutions $I^{\bullet \bullet}$?

4 - Can I then define the hypercohomology of a complex with respect to an arbitrary left exact functor, or does it have to always be the global section functor?

I say this because I want to apply the above theorem to obtain the hypercohomology of a cochain complex of sheaves $K^\bullet$ with respect to a composition of 2 left exact functors $G$ and $F$ $\mathbb{H}(G \circ F) (K^\bullet) = \mathbb{R}(G \circ F) (K^\bullet)$ (where $F$ is a global section functor and the other is a Hom-functor), when I know $\mathbb{H} F(K^\bullet)$. I wanted to know if I can proceed to use the theorem basically verbatim for hypercohomology or if there are other considerations I should know about.

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The theorem you cite is about derived categories and total derived functors. If you want to get anything concrete about cohomological derived functors you have to take a spectral sequence. – Zhen Lin Aug 14 '12 at 11:30

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