Profinite completion and double-dual : isomorphisms and counter-examples

TL;DR

This paper introduces a new approach to profinite completions of groups using finite approximations, proves an isomorphism between profinite completions and double-duals for vector spaces over finite fields, and discusses counter-examples in group theory.

Contribution

It provides a novel presentation of profinite completions and demonstrates a natural isomorphism between profinite completions and double-duals for vector spaces over finite fields.

Findings

Isomorphism between al{V} and V^{**} for finite field vector spaces

Counter-examples for iterated profinite completions in groups

Differences between algebraic and topological cases

Abstract

We define, for any group $G$, finite approximations ; with this tool, we give a new presentation of the profinite completion $\hat{\pi} : G \to \hat{G}$ of an abtract group $G$. We then prove the following theorem : if $k$ is a finite prime field and if $V$ is a $k$-vector space, then, there is a natural isomorphism between $\hat{V}$ (for the underlying additive group structure) and the additive group of the double-dual $V^{**}$. This theorem gives counter-examples concerning the iterated profinite completions of a group. These phenomena don't occur in the topological case.