A spectral sequence is a tool of homological algebra that has many applications in algebra, algebraic geometry, and algebraic topology. Roughly speaking, a spectral sequence is a system for keeping track of collections of exact sequences that have maps between them.
There are many definitions of spectral sequences and many slight variations that are useful for certain purposes. The most common type is a "first quadrant cohomological
spectral sequence," which is a collection of Abelian
groups 
where 
,
, and 
are integers, with 
and 
nonnegative and 
for some positive integer 
, usually 2. The groups 
come equipped with maps
 |
(1)
|
such that
 |
(2)
|
There is the further restriction that
 |
(3)
|
The maps 
are called boundary maps.
A spectral sequence may be visualized as a sequence of grids, one for each value of 
.
The 
s
and 
s
denote positions on the grid, where 
is the 
-coordinate and 
is the 
-coordinate. The diagram above shows this for 
.
The entire collection of groups 
together with their boundary maps is referred to as
the 
-term.
The groups 
are completely determined by the 
-term
For any given value of 
and 
, the group 
eventually stabilizes, because there are only a finite
number of nonzero boundary maps that either start or end at this position. This stable
value is referred to a 
. Stable values allow the convergence of a spectral
sequence to be defined. In particular, the spectral sequence 
converges to groups 
, written
 |
(4)
|
if there is a filtration
 |
(5)
|
such that the successive quotients are equal to the 
terms, i.e.,
 |
(6)
|
The Serre spectral sequence is used to compute the cohomology groups of the spaces in a fibration. Suppose
 |
(7)
|
is a fibration (for example, the Hopf map 
). Then there is a spectral sequence with
-term
 |
(8)
|
The sequence converges to 
. (Here, the 
denotes ordinary cohomology and is unrelated to the 
's above.) This allows one to compute
the cohomology of one of three spaces 
, 
, and 
from the cohomology of the
other two.
There are other examples of spectral sequences. The Leray-Serre spectral sequence is used to compute the hypercohomology of complexes of sheaves. The Grothendieck spectral sequence is used to compute the derived functors of the composition of two functors from the derived functors of the two original functors. The Leray-Serre spectral sequence is a special case of the Grothendieck spectral sequence. Finally, the Adams spectral sequence is used to compute the higher homotopy groups of spheres.
