Grothendieck spectral sequence

In mathematics, in the field of homological algebra, the Grothendieck spectral sequence, introduced by Alexander Grothendieck in his "Tôhoku" paper, is a spectral sequence that computes the derived functors of the composition of two functors formula_1, from knowledge of the derived functors of "F" and "G".
If formula_2 and formula_3 are two additive and left exact functors between abelian categories such that both formula_4 and formula_5 have enough injectives and formula_6 takes injective objects to formula_7-acyclic objects, then for each object formula_8 of formula_4 there is a spectral sequence:
where formula_11 denotes the "p"-th right-derived functor of formula_7, etc.
Many spectral sequences in algebraic geometry are instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.
The exact sequence of low degrees reads
Examples.
The Leray spectral sequence.
If formula_14 and formula_15 are topological spaces, let 
For a continuous map
there is the (left-exact) direct image functor
We also have the global section functors
and
Then since
and the functors 
formula_24 
and
formula_25 
satisfy the hypotheses (since the direct image functor has an exact left adjoint formula_26, pushforwards of injectives are injective and in particular acyclic for the global section functor), the sequence in this case becomes:
for a sheaf formula_28 of abelian groups on formula_14, and this is exactly the Leray spectral sequence.
Local-to-global Ext spectral sequence.
There is a spectral sequence relating the global Ext and the sheaf Ext: let "F", "G" be sheaves of modules over a ringed space formula_30; e.g., a scheme. Then
This is an instance of the Grothendieck spectral sequence: indeed,
Moreover, formula_35 sends injective formula_36-modules to flasque sheaves, which are formula_37-acyclic. Hence, the hypothesis is satisfied.
Derivation.
We shall use the following lemma:
Proof: Let formula_38 be the kernel and the image of formula_39. We have
which splits. This implies each formula_41 is injective. Next we look at
It splits, which implies the first part of the lemma, as well as the exactness of
Similarly we have (using the earlier splitting):
The second part now follows. formula_45
We now construct a spectral sequence. Let formula_46 be an "F"-acyclic resolution of "A". Writing formula_47 for formula_48, we have:
Take injective resolutions formula_50 and formula_51 of the first and the third nonzero terms. By the horseshoe lemma, their direct sum formula_52 is an injective resolution of formula_53. Hence, we found an injective resolution of the complex:
such that each row formula_55 satisfies the hypothesis of the lemma (cf. the Cartan–Eilenberg resolution.)
Now, the double complex formula_56 gives rise to two spectral sequences, horizontal and vertical, which we are now going to examine. On the one hand, by definition,
which is always zero unless "q" = 0 since formula_53 is "G"-acyclic by hypothesis. Hence, formula_59 and formula_60. On the other hand, by the definition and the lemma,
Since formula_62 is an injective resolution of formula_63 (it is a resolution since its cohomology is trivial),
Since formula_65 and formula_66 have the same limiting term, the proof is complete. formula_45