A chain complex is a sequence of maps
 |
(1)
|
where the spaces 
may be Abelian groups or modules.
The maps must satisfy 
.
Making the domain implicitly understood, the maps are denoted by 
, called the boundary
operator or the differential. Chain complexes are an algebraic tool for computing
or defining homology and have a variety of applications.
A cochain complex is used in the case of cohomology.
Elements of 
are called chains. For each 
, the kernel of 
is called the group of cycles,
 |
(2)
|
The letter