, in the field of homological algebra
, the Grothendieck spectral sequence
is a technique that allows one to compute the derived functors
of the composition of two functors
, from knowledge of the derived functors of F
are two additive (covariant) functors between abelian categories such that is left exact and takes injective objects of to -acyclic objects of , then there is a spectral sequence for each object of :
Many spectral sequences are merely instances of the Grothendieck spectral sequence, for example the Leray spectral sequence and the Lyndon-Hochschild-Serre spectral sequence.
The exact sequence of low degrees reads
- 0 → R1G(FA) → R1(GF)(A) → G(R1F(A)) → R2G(FA) → R2(GF)(A)
Example: the Leray spectral sequence
If and are topological spaces, let
- and be the category of sheaves of abelian groups on X and Y, respectively and
- be the category of abelian groups.
For a continuous map
there is the (left-exact) direct image functor
We also have the global section functors
and the functors
satisfy the hypotheses (injectives are flasque sheaves, direct images of flasque sheaves are flasque, and flasque sheaves are acyclic for the global section functor), the sequence in this case becomes:
for a sheaf of abelian groups on , and this is exactly the Leray spectral sequence.