- Source: Profinite integer
In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat)
Z
^
=
lim
←
Z
/
n
Z
=
∏
p
Z
p
{\displaystyle {\widehat {\mathbb {Z} }}=\varprojlim \mathbb {Z} /n\mathbb {Z} =\prod _{p}\mathbb {Z} _{p}}
where the inverse limit
lim
←
Z
/
n
Z
{\displaystyle \varprojlim \mathbb {Z} /n\mathbb {Z} }
indicates the profinite completion of
Z
{\displaystyle \mathbb {Z} }
, the index
p
{\displaystyle p}
runs over all prime numbers, and
Z
p
{\displaystyle \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
The profinite integers
Z
^
{\displaystyle {\widehat {\mathbb {Z} }}}
can be constructed as the set of sequences
υ
{\displaystyle \upsilon }
of residues represented as
υ
=
(
υ
1
mod
1
,
υ
2
mod
2
,
υ
3
mod
3
,
…
)
{\displaystyle \upsilon =(\upsilon _{1}{\bmod {1}},~\upsilon _{2}{\bmod {2}},~\upsilon _{3}{\bmod {3}},~\ldots )}
such that
m
|
n
⟹
υ
m
≡
υ
n
mod
m
{\displaystyle m\ |\ n\implies \upsilon _{m}\equiv \upsilon _{n}{\bmod {m}}}
.
Pointwise addition and multiplication make it a commutative ring.
The ring of integers embeds into the ring of profinite integers by the canonical injection:
η
:
Z
↪
Z
^
{\displaystyle \eta :\mathbb {Z} \hookrightarrow {\widehat {\mathbb {Z} }}}
where
n
↦
(
n
mod
1
,
n
mod
2
,
…
)
.
{\displaystyle n\mapsto (n{\bmod {1}},n{\bmod {2}},\dots ).}
It is canonical since it satisfies the universal property of profinite groups that, given any profinite group
H
{\displaystyle H}
and any group homomorphism
f
:
Z
→
H
{\displaystyle f:\mathbb {Z} \rightarrow H}
, there exists a unique continuous group homomorphism
g
:
Z
^
→
H
{\displaystyle g:{\widehat {\mathbb {Z} }}\rightarrow H}
with
f
=
g
η
{\displaystyle f=g\eta }
.
= Using Factorial number system
=Every integer
n
≥
0
{\displaystyle n\geq 0}
has a unique representation in the factorial number system as
n
=
∑
i
=
1
∞
c
i
i
!
with
c
i
∈
Z
{\displaystyle n=\sum _{i=1}^{\infty }c_{i}i!\qquad {\text{with }}c_{i}\in \mathbb {Z} }
where
0
≤
c
i
≤
i
{\displaystyle 0\leq c_{i}\leq i}
for every
i
{\displaystyle i}
, and only finitely many of
c
1
,
c
2
,
c
3
,
…
{\displaystyle c_{1},c_{2},c_{3},\ldots }
are nonzero.
Its factorial number representation can be written as
(
⋯
c
3
c
2
c
1
)
!
{\displaystyle (\cdots c_{3}c_{2}c_{1})_{!}}
.
In the same way, a profinite integer can be uniquely represented in the factorial number system as an infinite string
(
⋯
c
3
c
2
c
1
)
!
{\displaystyle (\cdots c_{3}c_{2}c_{1})_{!}}
, where each
c
i
{\displaystyle c_{i}}
is an integer satisfying
0
≤
c
i
≤
i
{\displaystyle 0\leq c_{i}\leq i}
.
The digits
c
1
,
c
2
,
c
3
,
…
,
c
k
−
1
{\displaystyle c_{1},c_{2},c_{3},\ldots ,c_{k-1}}
determine the value of the profinite integer mod
k
!
{\displaystyle k!}
. More specifically, there is a ring homomorphism
Z
^
→
Z
/
k
!
Z
{\displaystyle {\widehat {\mathbb {Z} }}\to \mathbb {Z} /k!\,\mathbb {Z} }
sending
(
⋯
c
3
c
2
c
1
)
!
↦
∑
i
=
1
k
−
1
c
i
i
!
mod
k
!
{\displaystyle (\cdots c_{3}c_{2}c_{1})_{!}\mapsto \sum _{i=1}^{k-1}c_{i}i!\mod k!}
The difference of a profinite integer from an integer is that the "finitely many nonzero digits" condition is dropped, allowing for its factorial number representation to have infinitely many nonzero digits.
= Using the Chinese Remainder theorem
=Another way to understand the construction of the profinite integers is by using the Chinese remainder theorem. Recall that for an integer
n
{\displaystyle n}
with prime factorization
n
=
p
1
a
1
⋯
p
k
a
k
{\displaystyle n=p_{1}^{a_{1}}\cdots p_{k}^{a_{k}}}
of non-repeating primes, there is a ring isomorphism
Z
/
n
≅
Z
/
p
1
a
1
×
⋯
×
Z
/
p
k
a
k
{\displaystyle \mathbb {Z} /n\cong \mathbb {Z} /p_{1}^{a_{1}}\times \cdots \times \mathbb {Z} /p_{k}^{a_{k}}}
from the theorem. Moreover, any surjection
Z
/
n
→
Z
/
m
{\displaystyle \mathbb {Z} /n\to \mathbb {Z} /m}
will just be a map on the underlying decompositions where there are induced surjections
Z
/
p
i
a
i
→
Z
/
p
i
b
i
{\displaystyle \mathbb {Z} /p_{i}^{a_{i}}\to \mathbb {Z} /p_{i}^{b_{i}}}
since we must have
a
i
≥
b
i
{\displaystyle a_{i}\geq b_{i}}
. It should be much clearer that under the inverse limit definition of the profinite integers, we have the isomorphism
Z
^
≅
∏
p
Z
p
{\displaystyle {\widehat {\mathbb {Z} }}\cong \prod _{p}\mathbb {Z} _{p}}
with the direct product of p-adic integers.
Explicitly, the isomorphism is
ϕ
:
∏
p
Z
p
→
Z
^
{\displaystyle \phi :\prod _{p}\mathbb {Z} _{p}\to {\widehat {\mathbb {Z} }}}
by
ϕ
(
(
n
2
,
n
3
,
n
5
,
⋯
)
)
(
k
)
=
∏
q
n
q
mod
k
{\displaystyle \phi ((n_{2},n_{3},n_{5},\cdots ))(k)=\prod _{q}n_{q}\mod k}
where
q
{\displaystyle q}
ranges over all prime-power factors
p
i
d
i
{\displaystyle p_{i}^{d_{i}}}
of
k
{\displaystyle k}
, that is,
k
=
∏
i
=
1
l
p
i
d
i
{\displaystyle k=\prod _{i=1}^{l}p_{i}^{d_{i}}}
for some different prime numbers
p
1
,
.
.
.
,
p
l
{\displaystyle p_{1},...,p_{l}}
.
Relations
= 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 direct product
Z
^
⊂
∏
n
=
1
∞
Z
/
n
Z
{\displaystyle {\widehat {\mathbb {Z} }}\subset \prod _{n=1}^{\infty }\mathbb {Z} /n\mathbb {Z} }
which is compact with its product topology by Tychonoff's theorem. Note the topology on each finite group
Z
/
n
Z
{\displaystyle \mathbb {Z} /n\mathbb {Z} }
is given as the discrete topology.
The topology on
Z
^
{\displaystyle {\widehat {\mathbb {Z} }}}
can be defined by the metric,
d
(
x
,
y
)
=
1
min
{
k
∈
Z
>
0
:
x
≢
y
mod
(
k
+
1
)
!
}
{\displaystyle d(x,y)={\frac {1}{\min\{k\in \mathbb {Z} _{>0}:x\not \equiv y{\bmod {(k+1)!}}\}}}}
Since addition of profinite integers is continuous,
Z
^
{\displaystyle {\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
Z
^
{\displaystyle {\widehat {\mathbb {Z} }}}
is the abelian group
Q
/
Z
{\displaystyle \mathbb {Q} /\mathbb {Z} }
equipped with the discrete topology (note that it is not the subset topology inherited from
R
/
Z
{\displaystyle \mathbb {R} /\mathbb {Z} }
, which is not discrete). The Pontryagin dual is explicitly constructed by the function
Q
/
Z
×
Z
^
→
U
(
1
)
,
(
q
,
a
)
↦
χ
(
q
a
)
{\displaystyle \mathbb {Q} /\mathbb {Z} \times {\widehat {\mathbb {Z} }}\to U(1),\,(q,a)\mapsto \chi (qa)}
where
χ
{\displaystyle \chi }
is the character of the adele (introduced below)
A
Q
,
f
{\displaystyle \mathbf {A} _{\mathbb {Q} ,f}}
induced by
Q
/
Z
→
U
(
1
)
,
α
↦
e
2
π
i
α
{\displaystyle \mathbb {Q} /\mathbb {Z} \to U(1),\,\alpha \mapsto e^{2\pi i\alpha }}
.
= Relation with adeles
=The tensor product
Z
^
⊗
Z
Q
{\displaystyle {\widehat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} }
is the ring of finite adeles
A
Q
,
f
=
∏
p
′
Q
p
{\displaystyle \mathbf {A} _{\mathbb {Q} ,f}={\prod _{p}}'\mathbb {Q} _{p}}
of
Q
{\displaystyle \mathbb {Q} }
where the symbol
′
{\displaystyle '}
means restricted product. That is, an element is a sequence that is integral except at a finite number of places. There is an isomorphism
A
Q
≅
R
×
(
Z
^
⊗
Z
Q
)
{\displaystyle \mathbf {A} _{\mathbb {Q} }\cong \mathbb {R} \times ({\hat {\mathbb {Z} }}\otimes _{\mathbb {Z} }\mathbb {Q} )}
= Applications in Galois theory and Etale homotopy theory
=For the algebraic closure
F
¯
q
{\displaystyle {\overline {\mathbf {F} }}_{q}}
of a finite field
F
q
{\displaystyle \mathbf {F} _{q}}
of order q, the Galois group can be computed explicitly. From the fact
Gal
(
F
q
n
/
F
q
)
≅
Z
/
n
Z
{\displaystyle {\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
F
q
{\displaystyle \mathbf {F} _{q}}
is given by the inverse limit of the groups
Z
/
n
Z
{\displaystyle \mathbb {Z} /n\mathbb {Z} }
, so its Galois group is isomorphic to the group of profinite integers
Gal
(
F
¯
q
/
F
q
)
≅
Z
^
{\displaystyle \operatorname {Gal} ({\overline {\mathbf {F} }}_{q}/\mathbf {F} _{q})\cong {\widehat {\mathbb {Z} }}}
which gives a computation of the absolute Galois group of a finite field.
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
π
1
e
t
(
X
)
{\displaystyle \pi _{1}^{et}(X)}
as the profinite completion of automorphisms
π
1
e
t
(
X
)
=
lim
i
∈
I
Aut
(
X
i
/
X
)
{\displaystyle \pi _{1}^{et}(X)=\lim _{i\in I}{\text{Aut}}(X_{i}/X)}
where
X
i
→
X
{\displaystyle X_{i}\to X}
is an Etale cover. Then, the profinite integers are isomorphic to the group
π
1
e
t
(
Spec
(
F
q
)
)
≅
Z
^
{\displaystyle \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
Z
^
↪
π
1
e
t
(
G
m
)
{\displaystyle {\hat {\mathbb {Z} }}\hookrightarrow \pi _{1}^{et}(\mathbb {G} _{m})}
since the covering maps come from the polynomial maps
(
⋅
)
n
:
G
m
→
G
m
{\displaystyle (\cdot )^{n}:\mathbb {G} _{m}\to \mathbb {G} _{m}}
from the map of commutative rings
f
:
Z
[
x
,
x
−
1
]
→
Z
[
x
,
x
−
1
]
{\displaystyle f:\mathbb {Z} [x,x^{-1}]\to \mathbb {Z} [x,x^{-1}]}
sending
x
↦
x
n
{\displaystyle x\mapsto x^{n}}
since
G
m
=
Spec
(
Z
[
x
,
x
−
1
]
)
{\displaystyle \mathbb {G} _{m}={\text{Spec}}(\mathbb {Z} [x,x^{-1}])}
. If the algebraic torus is considered over a field
k
{\displaystyle k}
, then the Etale fundamental group
π
1
e
t
(
G
m
/
Spec(k)
)
{\displaystyle \pi _{1}^{et}(\mathbb {G} _{m}/{\text{Spec(k)}})}
contains an action of
Gal
(
k
¯
/
k
)
{\displaystyle {\text{Gal}}({\overline {k}}/k)}
as well from the fundamental exact sequence in etale homotopy theory.
= Class field theory and the profinite integers
=Class field theory is a branch of algebraic number theory studying the abelian field extensions of a field. Given the global field
Q
{\displaystyle \mathbb {Q} }
, the abelianization of its absolute Galois group
Gal
(
Q
¯
/
Q
)
a
b
{\displaystyle {\text{Gal}}({\overline {\mathbb {Q} }}/\mathbb {Q} )^{ab}}
is intimately related to the associated ring of adeles
A
Q
{\displaystyle \mathbb {A} _{\mathbb {Q} }}
and the group of profinite integers. In particular, there is a map, called the Artin map
Ψ
Q
:
A
Q
×
/
Q
×
→
Gal
(
Q
¯
/
Q
)
a
b
{\displaystyle \Psi _{\mathbb {Q} }:\mathbb {A} _{\mathbb {Q} }^{\times }/\mathbb {Q} ^{\times }\to {\text{Gal}}({\overline {\mathbb {Q} }}/\mathbb {Q} )^{ab}}
which is an isomorphism. This quotient can be determined explicitly as
A
Q
×
/
Q
×
≅
(
R
×
Z
^
)
/
Z
=
lim
←
(
R
/
m
Z
)
=
lim
x
↦
x
m
S
1
=
Z
^
{\displaystyle {\begin{aligned}\mathbb {A} _{\mathbb {Q} }^{\times }/\mathbb {Q} ^{\times }&\cong (\mathbb {R} \times {\hat {\mathbb {Z} }})/\mathbb {Z} \\&={\underset {\leftarrow }{\lim }}\mathbb {(} {\mathbb {R} }/m\mathbb {Z} )\\&={\underset {x\mapsto x^{m}}{\lim }}S^{1}\\&={\hat {\mathbb {Z} }}\end{aligned}}}
giving the desired relation. There is an analogous statement for local class field theory since every finite abelian extension of
K
/
Q
p
{\displaystyle K/\mathbb {Q} _{p}}
is induced from a finite field extension
F
p
n
/
F
p
{\displaystyle \mathbb {F} _{p^{n}}/\mathbb {F} _{p}}
.
See also
p-adic number
Ring of adeles
Supernatural number
Notes
References
Connes, Alain; Consani, Caterina (2015). "Geometry of the arithmetic site". arXiv:1502.05580 [math.AG].
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/
https://euro-math-soc.eu/system/files/news/Hendrik%20Lenstra_Profinite%20number%20theory.pdf
Kata Kunci Pencarian:
- Profinite integer
- Profinite group
- Integer
- P-adic number
- Grothendieck's Galois theory
- Cyclic group
- Completion of a ring
- Stone space
- Arithmetic progression topologies
- Tensor product of modules
