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Let $A{\buildrel F\over\rightarrow}B{\buildrel G\over\rightarrow}C$ be additive functors between abelian categories.

Hartshorne, in Proposition 5.4 of Residues and Duality, constructs the obvious natural transformation $\zeta_{G,F}:R(GF)\Rightarrow (RG)(RF)$ and notes its obvious properties. He then remarks:

This proposition shows the convenience of derived functors in the context of derived categories. What used to be a spectral sequence becomes now simply a composition of functors. (And of course one can recover the old spectral sequence from this proposition by taking cohomology and using the spectral sequence of a double complex).

I've always felt like I'm missing something here. In what sense does the composition of functors (together with the natural transformation $\zeta_{G,F}$) replace the old spectral sequence? I understand that there's a conceptual insight here, but my question is thisquestions are these:

1. What (if anything) is an example of a statement that used to be proved by invoking the spectral sequence but can now be proved more succinctly using the composition of functors?

2. In what sense does the recovery of the spectral sequence actually use the derived formalism? Isn't the prescribed double complex exactly the same one I'd have written down if I'd never heard of derived categories?

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# Derived Functors Versus Spectral Sequences

Let $A{\buildrel F\over\rightarrow}B{\buildrel G\over\rightarrow}C$ be additive functors between abelian categories.

Hartshorne, in Proposition 5.4 of Residues and Duality, constructs the obvious natural transformation $\zeta_{G,F}:R(GF)\Rightarrow (RG)(RF)$ and notes its obvious properties. He then remarks:

This proposition shows the convenience of derived functors in the context of derived categories. What used to be a spectral sequence becomes now simply a composition of functors. (And of course one can recover the old spectral sequence from this proposition by taking cohomology and using the spectral sequence of a double complex).

I've always felt like I'm missing something here. In what sense does the composition of functors (together with the natural transformation $\zeta_{G,F}$) replace the old spectral sequence? I understand that there's a conceptual insight here, but my question is this:

What (if anything) is an example of a statement that used to be proved by invoking the spectral sequence but can now be proved more succinctly using the composition of functors?