In mathematics, 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 and G.
If
and
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 instances of the Grothendieck spectral sequence, for example the Leray spectral sequence.
The exact sequence of low degrees reads
If and 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 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 , and this is exactly the Leray spectral sequence.
This article incorporates material from Grothendieck spectral sequence on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.