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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeAug 27th 2014
   • (edited Aug 27th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI gave _[[Pontryagin duality for torsion abelian groups]]_ its own entry, cross-linked of course with the examples-section at _[[Pontryagin dual]]_ and with all other relevant entries. Mainly I wanted to record this diagram here in a way that one could link to it quasi-directly: $$ \array{ &\mathbb{Z}[p^{-1}]/\mathbb{Z} &\hookrightarrow& \mathbb{Q}/\mathbb{Z} &\hookrightarrow& \mathbb{R}/\mathbb{Z} \\ {}^{\mathllap{hom(-,\mathbb{R}/\mathbb{Z})}}\downarrow \\ &\mathbb{Z}_p &\leftarrow& \hat \mathbb{Z} &\leftarrow& \mathbb{Z} } $$

   I gave Pontryagin duality for torsion abelian groups its own entry, cross-linked of course with the examples-section at Pontryagin dual and with all other relevant entries.

   Mainly I wanted to record this diagram here in a way that one could link to it quasi-directly:

   $$\array{ &\mathbb{Z}[p^{-1}]/\mathbb{Z} &\hookrightarrow& \mathbb{Q}/\mathbb{Z} &\hookrightarrow& \mathbb{R}/\mathbb{Z} \\ {}^{\mathllap{hom(-,\mathbb{R}/\mathbb{Z})}}\downarrow \\ &\mathbb{Z}_p &\leftarrow& \hat \mathbb{Z} &\leftarrow& \mathbb{Z} }$$
2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 8th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexI notice that none of the three references currently given at *[[Pontryagin duality for torsion abelian groups]]* really seem to substantiate the main claims made in the entry, and only * Dinakar Ramakrishnan, Robert J. Valenza, Section 3 of: *Fourier Analysis on Number Fields*, Graduate Texts in Mathematics **186**, Springer 1999 ([doi:10.1007/978-1-4757-3085-2](https://link.springer.com/book/10.1007/978-1-4757-3085-2)) provides any relevant details at all. Maybe there is a more focused reference on the entry's topic?

   I notice that none of the three references currently given at Pontryagin duality for torsion abelian groups really seem to substantiate the main claims made in the entry, and only

   • Dinakar Ramakrishnan, Robert J. Valenza, Section 3 of: Fourier Analysis on Number Fields, Graduate Texts in Mathematics 186, Springer 1999 (doi:10.1007/978-1-4757-3085-2)

   provides any relevant details at all.

   Maybe there is a more focused reference on the entry’s topic?

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 8th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexfor when the editing functionality is back: A good reference is instead: * Luis Ribes, Pavel Zalesskii, Thm. 2.9.6 in: *Profinite Groups*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3. Folge: A Series of Modern Surveys in Mathematics), vol 40. Springer 2000 ([doi:10.1007/978-3-662-04097-3_2](https://doi.org/10.1007/978-3-662-04097-3_2))

   for when the editing functionality is back:

   A good reference is instead:

   • Luis Ribes, Pavel Zalesskii, Thm. 2.9.6 in: Profinite Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3. Folge: A Series of Modern Surveys in Mathematics), vol 40. Springer 2000 (doi:10.1007/978-3-662-04097-3_2)