Grothendieck spectral sequence
In mathematics, in the field of homological algebra, the Grothendieck spectral sequence is a spectral sequence that computes the derived functors of the composition of two functors , from knowledge of the derived functors of F and G.
Many spectral sequences are instances of the Grothendieck spectral sequence, for example the Leray 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 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:
- Weibel, Charles A. (1994), An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, ISBN 978-0-521-55987-4, OCLC 36131259, MR 1269324