Profinite integer: Difference between revisions

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The tensor product <math>\widehat{\mathbb{Z}} \otimes_{\mathbb{Z}} \mathbb{Q}</math> is the [[ring of finite adeles]] <math>\mathbf{A}_{\mathbb{Q}, f} = \prod_p{}^{'} \mathbb{Q}_p</math> of <math>\mathbb{Q}</math> where the prime ' means [[restricted product]].<ref>[ Questions on some maps involving rings of finite adeles and their unit groups].</ref>
The set of profinite integers has a topology in which it is a [[compact space|compact]] [[Hausdorff space]], coming from the fact that it can be seen as a closed subset of the product <math>\prod_n \mathbb{Z}/n\mathbb{Z}</math>, which is compact with its product topology by [[Tychonoff's theorem]]. Addition of profinite integers is continuous, so <math>\widehat{\mathbb{Z}}</math> becomes a compact Hausdorff abelian group, and thus its [[Pontryagin dual]] must be a discrete abelian group.
In fact the Pontryagin dual of <math>\widehat{\mathbb{Z}}</math> is the discrete abelian group <math>\mathbb{Q}/\mathbb{Z}</math>. This fact is exhibited by the pairing