If and are two additive and left exactfunctors 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.
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:
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 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:
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.
Godement, Roger (1973), Topologie algébrique et théorie des faisceaux, Paris: Hermann, MR0345092
Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. Vol. 38. Cambridge University Press. ISBN978-0-521-55987-4. MR1269324. OCLC36131259.
Computational Examples[]
Sharpe, Eric (2003). Lectures on D-branes and Sheaves (pages 18–19), arXiv:hep-th/0307245
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