The invertible elements of any commutative
topological ring are a group under multiplication.
In general is not a topological group if it is
endowed with the subset topology because inversion need
not be continuous (only multiplication and addition on
are required to be continuous). It is usual therefore
to give the following topology.
There is an injection

(21.1) 
of into the topological product . We give the
corresponding subset topology. Then with this topology is a
topological group and the inclusion map
is continous. To
see continuity of inclusion, note that this topology is finer (has at
least as many open sets) than the subset topology induced by
, since the projection maps
are
continuous.
Example 21.1.1
This is a ``nonexample''. The inverse map on
is continuous with
respect to the
adic topology. If
,
then
, so if
, then
Definition 21.1.2 (Idele Group)
The
of
is the group
of invertible
elements of the adele ring
.
We shall usually speak of
as a subset of , and will
have to distinguish between the
and topologies.
Example 21.1.3
For a rational prime
, let
be the adele whose
th
component is
and whose
th component, for
, is
.
Then
as
in
, for the following reason.
We must show that if
is a basic open set that contains the
adele
, the
for all sufficiently large
are contained in
. Since
contains
and is a basic
open set, it is of the form
where
if a finite set, and the
, for
, are
arbitrary open subsets of
that contain
.
If
is a prime larger than any prime in
, then
for
, is in
. This proves
convergence.
If the inverse map were continuous on
, then
the sequence of
would converge to
.
However, if
is an open set as above about
, then
for sufficiently large
,
none of the adeles
are
contained in
.
Lemma 21.1.4
The group of ideles
is the restricted topological project
of the with respect to the units
,
with the restricted product topology.
We omit the proof of Lemma 21.1.4, which is a
matter of thinking carefully about the definitions. The main
point is that inversion is continuous on for each .
(See Example 21.1.1.)
We have seen that is naturally embedded in , so
is naturally embedded in
.
Definition 21.1.5 (Principal Ideles)
We call
, considered as a subgroup of
, the
.
Lemma 21.1.6
The principal ideles are discrete as a subgroup of
.
Proof.
For
is discrete in
, so
is embedded in
by (
21.1.1) as a discrete subset.
(Alternatively, the subgroup topology on
is finer than the
topology coming from
being a subset of
, and
is
already discrete in
.)
Definition 21.1.7 (Content of an Idele)
The
of
is
Lemma 21.1.8
The map
is a continuous homomorphism of
the topological group
into
, where
we view
as a topological group under multiplication.
If is a number field, then is surjective.
Proof.
That the content map
satisfies the axioms of a homomorphisms
follows from the multiplicative nature of the defining formula
for
. For continuity, suppose
is an open interval
in
. Suppose
is such that
.
By considering small intervals about each nonunit component of
, we find an open neighborhood
of
such that
. It follows the
is open.
For surjectivity, use that each archimedean valuation is surjective,
and choose an idele that is at all but one archimedean valuation.
Remark 21.1.9
Note also that the
topology is that appropriate to a
group of operators on
: a basis of open sets
is the
, where
are, respectively,
compact and
open, and
consists of the
such that
and
.
Definition 21.1.10 (
Ideles)
The subgroup
of
is the subgroup of ideles
such that
. Thus
is the kernel of
, so we have an exact sequence
where the surjectivity on the right is only if
is a number field.
Lemma 21.1.11
The subset
of is closed as a subset,
and the subset topology on
coincides
with the
subset topology on
.
Proof.
Let
with
. To prove that
is closed in
, we find an
neighborhood
of
that does not meet
.
1st Case. Suppose that
(possibly ).
Then there is a finite set of such that
 contains all the with
, and

.
Then the set
can be
defined by
for sufficiently small
.
2nd Case. Suppose that
.
Then there is a finite set of such that
 contains all the with
, and
 if
an inequality
implies
(This is because for a nonarchimedean valuation, the
largest absolute value less than is , where is
the residue characteristic. Also, the upper bound in
Cassels's article is
instead of
,
but I think he got it wrong.)
We can choose
so small that
(for
) implies
Then
may be defined
by
This works because if
, then either
for all
, in
which case
, so
,
or
for some
, in
which case
so again
.
We next show that the
 and topologies on
are the same. If
, we must show that every
neighborhood of
contains an neighborhood
and viceversa.
Let
be an neighborhood of
. Then it
contains an neighborhood of the type
where
is a finite set of valuations
. This contains
the
neighborhood in which
in (
21.1.2)
is replaced by
.
Next let
be an
neighborhood. Then it contains
an
neighborhood of the form
where the finite set
contains at least all archimedean
valuations
and all valuations
with
. Since
, we may also
suppose that
is so small that (
21.1.4)
implies
Then the intersection of (
21.1.4) with
is the same as that of (
21.1.2)
with
, i.e., (
21.1.4)
defines an
neighborhood.
By the product formula we have that
.
The following result is of vital importance in class
field theory.
Proof.
After the preceeding lemma, it is enough to find
an
compact set
such that the map
is surjective. We take for
the set of
with
where
is any idele of content greater than
the
of Lemma
20.4.1.
Let
. Then the content of
equals
the content of
, so by Lemma 20.4.1
there is an such that
all $v$
Then
, as required.
Remark 21.1.13
The quotient
is totally disconnected in the function
field case. For the structure of its connected component in the
number field case, see papers of Artin and Weil in the ``Proceedings
of the Tokyo Symposium on Algebraic Number Theory, 1955'' (Science
Council of Japan) or [
AT90]. The determination of the
character group of
is global class field theory.
William Stein
20040506