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)
|
 |
(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 as 
.
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 over a simply connected CW-complex 
, with pathwise-connected
fiber 
(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.) More generally, for a pathwise-connected
base 
,
the coefficients 
form a local coefficient system determined
by the fibration. The displayed untwisted form applies
when the fundamental group of 
acts trivially on the fiber cohomology.
The Grothendieck spectral sequence computes the right derived functors of a composition of left exact functors between Abelian categories, assuming enough injectives and that the first functor sends injective objects to objects acyclic for the second. The Leray spectral sequence for the cohomology of sheaves is an example of this construction (The Stacks Project Authors 2026). The Serre spectral sequence above, also called the Leray-Serre spectral sequence, concerns a fibration. The classical Adams spectral sequence is used to compute the stable homotopy groups of spheres.
