In a chain complex of modules
 |
the module 
of 
-boundaries is the image of 
. It is a submodule of
and is contained in the module of
-cycles 
, which is the kernel of 
.
The complex is called exact at 
if 
.
In the chain complex
 |
where all boundary operators are the multiplication by 4, for all 
the module of 
-boundaries is 
, whereas the module of 
-cycles is 
.
