# nLab profinite completion of the integers

Contents

group theory

### Cohomology and Extensions

#### Compact objects

objects $d \in C$ such that $C(d,-)$ commutes with certain colimits

# Contents

## Definition

###### Definition

The profinite completion of the integers is the inverse limit

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

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 $p$

$\widehat{\mathbb{Z}} \simeq \underset{p\; prime}{\prod} \mathbb{Z}_p \,.$
###### 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.

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