Profinite integer

In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat)

${\widehat {\mathbb {Z} }}=\varprojlim \mathbb {Z} /n\mathbb {Z} =\prod _{p}\mathbb {Z} _{p}$

where

$\varprojlim \mathbb {Z} /n\mathbb {Z}$

indicates the profinite completion of $\mathbb {Z}$, the index $p$ runs over all prime numbers, and $\mathbb {Z} _{p}$ is the ring of p-adic integers. This group is important because of its relation to Galois theory, Étale homotopy theory, and the ring of Adeles. In addition, it provides a basic tractable example of a profinite group.

Construction and relations

Concretely the profinite integers will be the set of sequences $\upsilon$ such that $\upsilon (n)\in \mathbb {Z} /n\mathbb {Z}$ and $m\ |\ n\implies \upsilon (m)\equiv \upsilon (n){\bmod {m}}$. Pointwise addition and multiplication makes it a commutative ring. If a sequence of integers converges modulo n for every n then the limit will exist as a profinite integer. There is an embedding of the integers into the ring of profinite integers since there is the canonical injection

$\mathbb {Z} \hookrightarrow {\widehat {\mathbb {Z} }}$ where $n\mapsto (n{\bmod {1}},n{\bmod {2}},\dots ).$

Topological properties

The set of profinite integers has an induced topology in which it is a compact Hausdorff space, coming from the fact that it can be seen as a closed subset of the infinite product

${\hat {\mathbb {Z} }}\subset \prod _{n}\mathbb {Z} /n\mathbb {Z}$

which is compact with its product topology by Tychonoff's theorem. Note the topology on each finite group $\mathbb {Z} /n\mathbb {Z}$ is given as the discrete topology. Since addition of profinite integers is continuous, ${\widehat {\mathbb {Z} }}$ is a compact Hausdorff abelian group, and thus its Pontryagin dual must be a discrete abelian group. In fact the Pontryagin dual of ${\widehat {\mathbb {Z} }}$ is the discrete abelian group $\mathbb {Q} /\mathbb {Z}$. This fact is exhibited by the pairing

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

where $\chi$ is the character of $\mathbf {A} _{\mathbb {Q} ,f}$ induced by $\mathbb {Q} /\mathbb {Z} \to U(1),\,\alpha \mapsto e^{2\pi i\alpha }$.

Relation with adeles

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 symbol $'$ means restricted product.

Applications in Galois theory and Etale homotopy theory

For the algebraic closure ${\overline {\mathbf {F} }}_{q}$ of a finite field $\mathbf {F} _{q}$ of order q, the Galois group can be computed explicitly. From the fact ${\text{Gal}}(\mathbf {F} _{q^{n}}/\mathbf {F} _{q})\cong (\mathbb {Z} /n\mathbb {Z} )^{*}$ where the automorphisms are given by the Frobenius endomorphism, the Galois group of the algebraic closure of $\mathbf {F} _{q}$ is given by a profinite tower from the groups $(\mathbb {Z} /n\mathbb {Z} )^{*}$, it's Galois group is isomorphic to the group of profinite integers, hence

$\operatorname {Gal} ({\overline {\mathbf {F} }}_{q}/\mathbf {F} _{q})={\widehat {\mathbb {Z} }}$

Relation with Etale fundamental groups of algebraic tori

This construction can be re-interpreted in many ways. One of them is from Etale homotopy theory which defines the Etale fundamental group $\pi _{1}^{et}(X)$ as the profinite completion of automorphisms

$\pi _{1}^{et}(X)=\lim _{i\in I}{\text{Aut}}(X_{i}/X)$

where $X_{i}\to X$ is an Etale cover. Then, the profinite integers are isomorphic to the group

$\pi _{1}^{et}({\text{Spec}}(\mathbf {F} _{q}))\cong {\hat {\mathbb {Z} }}$

from the earlier computation of the profinite Galois group. In addition, there is an embedding of the profinite integers inside the Etale fundamental group of the algebraic torus

${\hat {\mathbb {Z} }}\hookrightarrow \pi _{1}^{et}(\mathbb {G} _{m})$

since the covering maps come from the polynomial maps

$(\cdot )^{n}:\mathbb {G} _{m}\to \mathbb {G} _{m}$

from the map of commutative rings

$f:\mathbb {Z} [x,x^{-1}]\to \mathbb {Z} [x,x^{-1}]$ sending $x\mapsto x^{n}$

since $\mathbb {G} _{m}={\text{Spec}}(\mathbb {Z} [x,x^{-1}])$. If the algebraic torus is considered over a field $k$, then the Etale fundamental group $\pi _{1}^{et}(\mathbb {G} _{m}/{\text{Spec(k)}})$ contains an action of ${\text{Gal}}({\overline {k}}/k)$ as well from the fundamental exact sequence in etale homotopy theory.

See also

• p-adic number
• Supernatural number

References

• Connes, Alain; Consani, Caterina (2015). "Geometry of the arithmetic site". arXiv:1502.05580.
• Milne, J.S. (2013-03-23). "Class Field Theory" (PDF). Archived from the original (PDF) on 2013-06-19. Retrieved 2020-06-07.

External links

• http://ncatlab.org/nlab/show/profinite+completion+of+the+integers
• https://web.archive.org/web/20150401092904/http://www.noncommutative.org/supernatural-numbers-and-adeles/