What is a spectral sequence?

A (cohomological) spectral sequence has

pages , where is an integer (possibly or ).

Each page contains:

an abelian group for each (often, unless ); and
a differential ().

To get to the next page,

taken at :

This doesn’t give a recipe for .

Sometimes (often?) it converges.

Definition (Convergence)

For all , there exists such that for all ,

(because at ). We write the common value of the group to be

If the spectral sequence is a first-quadrant spectral sequence, i.e., unless , then we always have convergence. To see this, note that:

if then so ,
if then so .

What does the spectral sequence converge to?

#TODO2
Under the assumption that the spectral sequence is first-quadrant, the spectral sequence converges to (abelian groups) for all if for all we have a filtration

with

In this case, we write that . #TODO diagram

Example: Lyndon-Hochschild-Serre spectral sequence

#TODO2
Let be a group. Let be a normal subgroup. Let be a -module. There is a spectral sequence

That is to say, is built from the -page in the above above way.

Example: Serre spectral sequence

Let be a fibration, e.g., a fibre bundle where locally on , there exists an open cover such that

Assume that is simply-connected, i.e., . Then there is a spectral sequence

Example: Hopf fibration

We have

There is a map to

where a line means a -(complex) dimensional subspace:

What is the fibre of , i.e., when is ? Suppose that we have

So and . Then

Therefore . So we are forced to have . Hence, the fibre here is .

This fibre bundle is not a product: it is well known that .

Cohomology of

For ,

Spectral sequence

#TODO2
We have

Example

#TODO2
Consider the flag variety

There is a projection down to

The fibre is

The cohomology of is given by

So the spectral sequence is

We find that

, ,
, ,
and so on.

In fact, because the spectral sequence is only supported on even coordinates , , and the differential goes down and right , we must have for all by a parity argument. We say that the spectral sequence degenerates at .

We calculate that

there exists a short exact sequence

2024-09-04