Today we’ll start a new section, but only because it is a tool we need when we come back to the stuff we just finished. We will look at completions.
To motivate the process take a Hausdorff abelian topological group . Suppose there is a countable local basis at 0 (which implies countable basis, since the neighborhoods of 0 determine the entire topology). Since we assumed Hausdorff we have the usual notion of Cauchy sequences, so we can define the completion of to be completion in the usual sense . In particular, if , then .
Now suppose we have a local basis about 0 of subgroups (this rules out ), say . If we are in this situation, then our topology is actually determined by a sequence of subgroups, so we will want to try to define the completion solely in terms of algebra.
Take any Cauchy sequence . If we fix k, then at some , is constant for all . Note that really does depend on . Set the limit .
If we make what we mod out by bigger, namely we go from to , then projection maps . Thus our Cauchy sequence determined a “coherent sequence” , where .
Conversely, we can define a Cauchy sequence corresponding to any coherent sequence by just picking an element in the equivalence class at each step. So we can now define the completion to be the set of coherent sequences with group structure given entry-wise by the quotient group. The standard example here is the -adic integers, where the group is and our fundamental system is . Coherent sequences are where .
Whenever we have in general a sequence of groups and homomorphisms this is called an inverse system. The group of all coherent sequences is called the inverse limit of the system written . Thus our definition of completion can be written succinctly as .
Next time we’ll transfer this to module language and get to a few results.