A coboundary in a cochain complex is an element of the image of its preceding differential. For a cochain complex of modules
 |
the module 
of 
-coboundaries
is the image of 
. It is a submodule of
and is contained in the module
of 
-cocycles 
.
The cochain complex is an exact sequence at 
if 
.
In the right cochain complex of 
-modules
 |
for all 
, the 
th module is 
, and the 
th coboundary operator
maps every element 
of 
to the residue class of 
in 
. The module of 
-coboundaries consists of the residue
classes of 0 and 
in 
for 
, while 
. For every 
, the module of 
-cocycles consists of the residue
classes of all even numbers 0, 2, 4, ..., 
.
