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profinite completion of the integers

Contents

Context

Group Theory

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Compact objects

Contents

Definition

Definition

The profinite completion of the integers is the inverse limit

^lim n(/n) \widehat{\mathbb{Z}} \coloneqq \underset{\leftarrow}{\lim}_{n \in \mathbb{N}} (\mathbb{Z}/n\mathbb{Z})

of all the cyclic groups over their canonical filtered diagram.

Proposition

This is isomorphic to the product of the p-adic integers for all prime numbers pp

^pprime p. \widehat{\mathbb{Z}} \simeq \underset{p\; prime}{\prod} \mathbb{Z}_p \,.
Remark

From this perspective the concept of the ring of adeles is natural, see there for more.

Properties

Pontryagin duality

Under Pontryagin duality, ^\hat \mathbb{Z} maps to /\mathbb{Q}/\mathbb{Z}, see at Pontryagin duality for torsion abelian groups.

[p 1]/ / / hom(,/) p ^ \array{ &\mathbb{Z}[p^{-1}]/\mathbb{Z} &\hookrightarrow& \mathbb{Q}/\mathbb{Z} &\hookrightarrow& \mathbb{R}/\mathbb{Z} \\ {}^{\mathllap{hom(-,\mathbb{R}/\mathbb{Z})}}\downarrow \\ &\mathbb{Z}_p &\leftarrow& \hat \mathbb{Z} &\leftarrow& \mathbb{Z} }

References

Last revised on August 26, 2014 at 23:08:55. See the history of this page for a list of all contributions to it.