# 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.