nLab Pontryagin duality for torsion abelian groups



Group Theory




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

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

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


  • /\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 p\mathbb{Z}_p is Pontryagin dual to the Prüfer p-group [p 1]/\mathbb{Z}[p^{-1}]/\mathbb{Z}.

[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} }


  • 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)

Last revised on January 22, 2020 at 12:15:00. See the history of this page for a list of all contributions to it.