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