Grothendieck spectral sequence
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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 G is left exact and F takes injective objects of 
to G-acyclic objects of 
, then there is a spectral sequence for each object A of :
[edit] Example: the Leray spectral sequence
If X and Y are topological spaces, let
be the category of sheaves of abelian groups on X,
be the category of sheaves of abelian groups on Y
and
be the category of abelian groups.
Then for a continuous map
we have a functor
,
the direct image functor. We also have the global section functors
,
and
Then since
and the functors f * and ΓY 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 X, and this is exactly the Leray spectral sequence.
[edit] See also
[edit] References
- An introduction to homological algebra, Weibel
This article incorporates material from Grothendieck spectral sequence on PlanetMath, which is licensed under the GFDL.
