Grothendieck Spectral Sequence - Example: The Leray Spectral Sequence

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.