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
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.
Read more about this topic: Grothendieck Spectral Sequence
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