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, from knowledge of the derived functors of F and G.
If and are two additive and left exact functors between abelian categories such that both and have enough injectives and takes injective objects to -acyclic objects, then for each object of there is a spectral sequence:
where denotes the p-th right-derived functor of, 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 and are topological spaces, let
For a continuous map
there is the direct image functor
We also have the global section functors
and
Then since
and the functors
and
satisfy the hypotheses, the sequence in this case becomes:
for a sheaf of abelian groups on, 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 ; e.g., a scheme. Then
This is an instance of the Grothendieck spectral sequence: indeed,
Moreover, sends injective -modules to flasque sheaves, which are -acyclic. Hence, the hypothesis is satisfied.

Derivation

We shall use the following lemma:
Proof: Let be the kernel and the image of. We have
which splits. This implies each 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 :
The second part now follows.
We now construct a spectral sequence. Let be an F-acyclic resolution of A. Writing for, we have:
Take injective resolutions and of the first and the third nonzero terms. By the horseshoe lemma, their direct sum is an injective resolution of. Hence, we found an injective resolution of the complex:
such that each row satisfies the hypothesis of the lemma
Now, the double complex 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 is G-acyclic by hypothesis. Hence, and. On the other hand, by the definition and the lemma,
Since is an injective resolution of ,
Since and have the same limiting term, the proof is complete.
Computational Examples