The inverse limit of a family of 
-modules is the dual notion of a direct
limit and is characterized by the following mapping property. For a directed
set 
and a family of 
-modules
, let 
be an inverse
system. 
is some 
-module
with some homomorphisms 
,
where for each 
,
 |
(1)
|
such that if there exists some 
-module 
with homomorphisms 
, where for each 
, 
 |
(2)
|
then a unique homomorphism 
is induced and the above diagram
commutes.
The inverse limit can be constructed as follows. For a given inverse system, 
,
write
 |
(3)
|
