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.
If
and
are two additive and left exact(covariant) functors between abelian categories such that 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
- 0 → R1G(FA) → R1(GF)(A) → G(R1F(A)) → R2G(FA) → R2(GF)(A).
Read more about Grothendieck Spectral Sequence: Example: The Leray Spectral Sequence
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