In 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 called exact at 
if 
.
In the right 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 is the set of the residue classes of 0 and 
in 
, and the module of 
-cocycles the set of the residues
classes of all even numbers 
.
