A chain complex is a sequence of group homomorphisms or module homomorphisms
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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 group kernel of 
is called the
group of cycles,
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The letter 
is short for the German word for cycle, "Zyklus." The image 
is contained in the group of cycles because
. It is called
the group of boundaries.
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The quotients 
are the homology groups
of the chain complex.
For example, the sequence
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where every space is 
and each map is given by multiplication
by 4 is a chain complex. The cycles at each stage
are 
and the boundaries are 
. So the homology at each
stage is the group of two elements 
. A simpler example is given by a
linear transformation 
, which can be extended to a chain complex by the
zero vector space and the
zero map. Then the nontrivial homology
groups are 
and 
.
The terminology of chain complexes comes from the calculation for homology of geometric objects in a topological space,
like a manifold. For example, in the figure above, let
and 
denote the points, and 
and 
denote the oriented segments,
which are the chains. The boundary of 
is 
, and the boundary of
is 
.
The group 
is the free Abelian group 
and the group 
is the free
Abelian group 
.
The boundary operator is
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The other groups 
are the trivial group,
and the other maps are the zero map.
Then 
is generated by 
and 
is the trivial subgroup. So 
is the rank one free
Abelian group isomorphic to 
. The zero-dimensional
case is slightly more interesting. Every element of 
has no boundary
and so is in 
while the boundaries 
are generated by 
. Hence, 
is also isomorphic
to 
. Note that the result is not affected
by how the circle is cut into pieces, or by how many cuts
are used.
