Profinite integer

In mathematics, a profinite integer is an element of the ring

\widehat {{\mathbb {Z}}}=\prod _{p}{\mathbb {Z}}_{p}

where p runs over all prime numbers, $\mathbb {Z} _{p}$ is the ring of p-adic integers and ${\widehat {\mathbb {Z} }}=\varprojlim \mathbb {Z} /n\mathbb {Z}$ (profinite completion).

Example: Let $\overline {{\mathbf {F}}}_{q}$ be the algebraic closure of a finite field $\mathbf{F}_q$ of order q. Then $\operatorname {Gal}(\overline {{\mathbf {F}}}_{q}/{\mathbf {F}}_{q})=\widehat {{\mathbb {Z}}}$ .^[1]

A usual (rational) integer is a profinite integer since there is the canonical injection

{\mathbb {Z}}\hookrightarrow \widehat {{\mathbb {Z}}},\,n\mapsto (n,n,\dots ).

The tensor product $\widehat {{\mathbb {Z}}}\otimes _{{{\mathbb {Z}}}}{\mathbb {Q}}$ is the ring of finite adeles ${\mathbf {A}}_{{{\mathbb {Q}},f}}=\prod _{p}{}^{{'}}{\mathbb {Q}}_{p}$ of $\mathbb {Q}$ where the prime ' means restricted product.^[2]

There is a canonical paring

{\mathbb {Q}}/{\mathbb {Z}}\times \widehat {{\mathbb {Z}}}\to U(1),\,(q,a)\mapsto \chi (qa)

^[3]

where $\chi$ is the character of ${\mathbf {A}}_{{{\mathbb {Q}},f}}$ induced by ${\mathbb {Q}}/{\mathbb {Z}}\to U(1),\,\alpha \mapsto e^{{2\pi i\alpha }}$ .^[4] The pairing identifies ${\widehat {\mathbb {Z} }}$ with the Pontrjagin dual of ${\mathbb {Q}}/{\mathbb {Z}}$ .

Notes

↑ Milne, Ch. I Example A. 5.
↑ http://math.stackexchange.com/questions/233136/questions-on-some-maps-involving-rings-of-finite-adeles-and-their-unit-groups
↑ Connes–Consani, § 2.4.
↑ K. Conrad, The character group of Q

References

Connes, Alain; Consani, Caterina (2015). "Geometry of the arithmetic site". arXiv:1502.05580 .
Milne, Class Field Theory

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Profinite integer

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Notes

References

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