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 an 
-module with module homomorphisms 
, where for all 
with 
 |
(1)
|
such that if there exists an 
-module 
with module homomorphisms 
, where for all 
with 
 |
(2)
|
then a unique module homomorphism 
is induced with 
for every 
,
so the above diagram commutes.
The inverse limit can be constructed as follows. For a given inverse system, 
,
write
 |
(3)
|
