## Generalities on completionsAugust 23, 2009

Posted by Akhil Mathew in algebra, commutative algebra, topology.
Tags: , ,

Today I’ll discuss completions in their algebraic context. All this is really a version of Cauchy’s construction of the real numbers, but it’s also useful in algebra, since one can study a ring through its completions (e.g. in algebraic number theory, as I hope to get to soon).

Generalities on Completions

Suppose we have a filtered abelian group ${G}$ with a descending filtration of subgroups ${\{G_i\}}$. Because of this, we can consider “Cauchy sequences” and “convergence:”

Definition 1

The sequence ${\{x_i\} \subset G}$, ${i \in \mathbb{N}}$ is Cauchy if for each ${A}$, there exists ${N}$ large enough that

$\displaystyle i,j > N \quad \mathrm{implies} \quad x_i - x_j \in G_A.$

The sequence ${\{y_i\} \subset G}$ converges to ${y}$ if for each ${A}$, there exists ${N}$ large enough that

$\displaystyle i>A \quad \mathrm{implies} \quad x_i -y \in G_A.$ (more…)