nLab Pontryagin duality for torsion abelian groups

Statement

Pontryagin duality restricts to an equivalence of categories between abelian profinite groups and (the opposite of) abelian torsion groups

Notice that for $G$ a profinite group then $hom(G,\mathbb{R}/\mathbb{Z})\simeq hom(G,\mathbb{Q}/\mathbb{Z})$ and so one usually writes this equivalence as

$$hom(-,\mathbb{Q}/\mathbb{Z}) \;\colon\; ProFinAb \stackrel{\simeq}{\longrightarrow} TorsionAb^{op}$$

Examples

• $\mathbb{Q}/\mathbb{Z}$ is dual to the profinite completion of the integers $\hat \mathbb{Z}$. And so the canonical map $\mathbb{Z} \longrightarrow \hat \mathbb{Z}$ is dual to $\mathbb{Q}/\mathbb{Z} \longrightarrow \mathbb{R}/\mathbb{Z}$.

• The abelian group underlying the p-adic integers $\mathbb{Z}_p$ is Pontryagin dual to the Prüfer p-group $\mathbb{Z}[p^{-1}]/\mathbb{Z}$.

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

References

• Jean-Pierre Serre, section 1.1. of Galois cohomology

• Ramakrishnan, Valenza, Fourier analysis on number fields

• Clark Barwick, Exercises on locally compact abelian groups: An invitation to harmonic analysis (pdf)