We will call a topological group complete if is an isomorphism.
The case that we are particularly concerned with is when our group is a ring and we take for our inverse system some ideal and . The topology that this determines is the “-adic topology”. This makes into a topological ring.
If we take the completion , then the continuous ring homomorphism has kernel .
Now we can also do all this with -modules by taking the group to be and the inverse system . The topology determined by this system is called the -topology on M. If we take the completion with respect to this topology (i.e. w.r.t this system), we get which is a topological -module meaning the action is continuous.
Rephrasing the motivating example from last time in this language we see that the -adic integers are formed as the completion of the ring with respect to the -topology where is the ideal .
The other really important example is to form the completion of with respect to the -adic topology. The completion is the ring of formal power series. Recall that by definition the inverse limit are all sequences such that . This just says that each is a polynomial, and it has to agree with the one before it up to the coefficient. So we can write each sequence where is the coefficient on the of the polynomial . And for any power series we get a sequence in this way.
Recall our notion of -filtrations. We had a chain such that , and if equality held for all large , then we called the filtration stable. Well, in our new language, these filtrations are inverse systems of modules, and hence determine a topology on . A few posts ago we used the fact that any stable -filtrations have bounded difference. In this new language, this says precisely that all stable -filtrations determine the same topology on M, moreover this is the -topology.
Lastly, if we convert the Artin-Rees Lemma to this language, we get that if is Noetherian, an ideal, a f.g. -module, and a submodule of , then the -topology on is actually just the subspace topology from the -topology on .
We should probably do some properties of completions next time.