Completions I

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 G. 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 G to be completion in the usual sense \hat{G}. In particular, if G=\mathbb{Q}, then \hat{\mathbb{Q}}=\mathbb{R}.

Now suppose we have a local basis about 0 of subgroups (this rules out \mathbb{Q}), say G=G_0\supset G_1\supset \cdots \supset G_n\supset \cdots . 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 (x_n)\subset G. If we fix k, then at some M(k), \overline{x_n}\in G/G_k is constant for all n\geq M(k). Note that M really does depend on k. Set the limit \overline{x_n}\to x_{M(k)}.

If we make what we mod out by bigger, namely we go from k+1 to k, then projection \theta_{k+1}: G/G_{k+1}\to G/G_k maps x_{M(k+1)}\mapsto x_{M(k)}. Thus our Cauchy sequence (x_n) determined a “coherent sequence” (x_{M(k)}), where \theta_{n+1}x_{M(n+1)}=x_{M(n)}.

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 \hat{G} to be the set of coherent sequences with group structure given entry-wise by the quotient group. The standard example here is the p-adic integers, where the group is \mathbb{Z} and our fundamental system is \mathbb{Z}\supset p\mathbb{Z}\supset p^2\mathbb{Z}\supset \cdots \supset p^n\mathbb{Z}\supset \cdots . Coherent sequences are (a_0, a_1, \ldots ) where a_{n+1}\mod p^n\equiv a_n.

Whenever we have in general a sequence of groups \{A_n\} and homomorphisms \theta_{n+1} A_{n+1}\to A_n this is called an inverse system. The group of all coherent sequences is called the inverse limit of the system written \displaystyle \lim_{\longleftarrow} A_n. Thus our definition of completion can be written succinctly as \displaystyle\hat{G}=\lim_{\longleftarrow} G/G_n.

Next time we’ll transfer this to module language and get to a few results.


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