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) |
