nLab profinite completion of the integers

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.

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

Extension of $U(1)$

There exists an extension of $U(1)$ by the profinite integers $\widehat{\mathbb{Z}}$

$$1 \to \widehat{\mathbb{Z}} \to S^1_{\mathbb{Q}} \to U(1) \to 1,$$

called the universal one–dimensional solenoid or adelic solenoid $S^1_{\mathbb{Q}}$, which can be thought of as the limit of the $p$-fold covers of $U(1)=S^1$.

This extension is equivalently the quotient of the adèle group of the rational numbers $A_{\mathbb{Q}}$ by $\mathbb{Q}$

$$S^1_{\mathbb{Q}} = A_{\mathbb{Q}} / \mathbb{Q}.$$

See (Ramakrishnan and Valenza ‘99), and (Burgos and Verjovsky ‘16) for more.

Related concepts

• the modular group/general linear group with profinite integer coefficients $GL_2(\widehat{\mathbb{Z}})$ appears as the symmetry group in modular equivariant elliptic cohomology.

References

• Hendrik Lenstra, example 2.2 in Profinite groups (pdf)

• Groupprops, Profinite completion of the integers

• Dinakar Ramakrishnan, Robert Valenza, Fourier Analysis on Number Fields. Graduate Texts in Mathematics 186, Springer-Verlag, New York, (1999). (doi).

• Juan Manuel Burgos, Alberto Verjovsky. Adelic solenoid I: Structure and topology (2016). (arXiv:1603.05676).

In the context of gauge theory:

• Pavel Putrov: $\mathbb{Q}/\mathbb{Z}$ symmetry [arXiv:2208.12071]