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 and .
Many spectral sequences in
algebraic geometry are instances of the Grothendieck spectral sequence, for example the
Leray spectral sequence.
Statement
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., and where the arrow '' means
convergence of spectral sequences.
Five term exact sequence
The
exact sequence of low degrees reads
Examples
The Leray spectral sequence
If and are
topological spaces, let and be the
category of sheaves of abelian groups on and , respectively.
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 and satisfy the hypotheses (since the direct image functor has an exact left adjoint , pushforwards of injectives are injective and in particular
acyclic for the global section functor), the
sequence in this case becomes:
for a
sheaf of abelian groups on .
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
-
[1]
This is an instance of the Grothendieck spectral sequence: indeed,
- , and .
Moreover, sends injective -modules to flasque sheaves,
[2] which are -acyclic. Hence, the hypothesis is satisfied.
Derivation
We shall use the following lemma:
Lemma — If K is an injective complex in an abelian category C such that the kernels of the differentials are injective objects, then for each n,
is an injective object and for any left-exact additive functor G on C,
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 (using the earlier splitting):
The second part now follows.
We now construct a spectral sequence. Let be an injective 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 (cf. the
Cartan–Eilenberg resolution.)
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 (it is a resolution since its cohomology is trivial),
Since and have the same limiting term, the proof is complete.
Notes
References
Computational Examples
This article incorporates material from Grothendieck spectral sequence on
PlanetMath, which is licensed under the
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