I'm trying to understand how derived categories "replace" spectral sequences. More specifically the derived category statement of Grothendieck Spectral Sequence vs the normal version. I been reading mostly in Weibel as well as this mathoverflow post https://mathoverflow.net/questions/91785/derived-functors-versus-spectral-sequences.
As far as I understand the picture is the following. When first doing Grothendieck spectral sequence one proofs it by using the theory of hyper(co)homology and the associated spectral sequences we get from Cartan-Eilenberg resolutions. This version I feel is understandable.
Here comes the part which I don't really understand. The Grothendieck spectral sequence in the derived category setting is "just" that there is an natural isomorphism $$ \textbf{R}F \circ \textbf{R}G \cong \textbf{R}(F \circ G)$$ which seems to be fine. What I dont understand is how we can recover the spectral sequence from this statement? Is there some naturally associated bicomplex? I know that if we take the homology of the derived functors then we get the hyperderived functors, but as said I don't understand how one gets a spectral sequence.
Thank you very much for any help! :)
One perspective on this is that derived categories are interesting, and one of the yardsticks we can use to measure an object is by taking its cohomology. However we have other ways of measuring our objects, such as homs to or from other objects, these still yield long exact sequences , since the derived category is triangulated. From this perspective, the composition of derived functors statement is a fundamental fact about going between different categories of objects, and is relevant to more than just cohomology, for instance, if we have adjoints, we can compute things more easily, they work for other cohomological functors, etc. The Grothendieck spectral sequence arises from taking the composition isomorphism, and “manually evaluating” its cohomology. By definition of derived functors (assuming enough resolving objects) we resolve an object X, apply G to it to get our object representing RG(X). Then resolve RG(X), apply F to get an object representing RF(RG(X)). By our comparison theorem, this object has cohomology of R(FG)(X). But if we started with an object X in our abelian category, then we can resolve (pick a quasi-iso) to a chain complex, then apply G, the resolve this new complex vertically, (and take total complex) then apply F. So the spectral sequence falls out of the fact that we resolve twice, we get a double complex then it’s the usual spectral sequence of a double complex. From this perspective, lots of other spectral sequences are “obvious” in the sense that they are just what you get if you evaluate the higher composition laws on stuff.