1Basic theory

III Local Fields 1.3 Topological rings
Recall that we previously constructed the valuation ring
O
K
. Since the valued
field
K
itself has a topology, the valuation ring inherits a subspace topology.
This is in fact a ring topology.
Definition
(Topological ring)
.
Let
R
be a ring. A topology on
R
is called a
ring topology if addition and multiplication are continuous maps
R × R R
. A
ring with a ring topology is a topological ring.
Example. R
and
C
with the usual topologies and usual ring structures are
topological rings.
Exercise.
Let
K
be a valued field. Then
K
is a topological ring. We can see
this from the fact that the product topology on
K ×K
is induced by the metric
d((x
0
, y
0
), (x
1
, y
1
)) = max(|x
0
x
1
|, |y
0
y
1
|).
Now if we are just randomly given a ring, there is a general way of constructing
a ring topology. The idea is that we pick an ideal
I
and declare its elements to
be small. For example, in a valued ring, we can pick
I
=
{x O
K
:
|x| <
1
}
.
Now if you are not only in
I
, but
I
2
, then you are even smaller. So we have a
hierarchy of small sets
I I
2
I
3
I
4
···
Now to make this a topology on
R
, we say that a subset
U R
is open if every
x U
is contained in some translation of
I
n
(for some
n
). In other words, we
need some y R such that
x y + I
n
U.
But since
I
n
is additively closed, this is equivalent to saying
x
+
I
n
U
. So we
make the following definition:
Definition
(
I
.
Let
R
be a ring and
I R
an ideal. A subset
U R
is called
I
x U
, there is some
n
1 such that
x + I
n
U.
Proposition.
The set of all
I
-adically open sets form a topology on
R
, called
Note that the
I
-adic topology isn’t really the kind of topology we are used
to thinking about, just like the topology on a valued field is also very weird.
Instead, it is a “filter” for telling us how small things are.
Proof.
By definition, we have
and
R
are open, and arbitrary unions are clearly
open. If
U, V
are
I
x U V
, then there are
n, m
such that
x + I
n
U and x + I
m
V . Then x + I
max(m,n)
U V .
Exercise. Check that the I-adic topology is a ring topology.
In the special case where
I
=
xR
, we often call the
I
x
topology.
Now we want to tackle the notion of completeness. We will consider the case
of I = xR for motivation, but the actual definition will be completely general.
If we pick the
x
-adic topology, then we are essentially declaring that we take
x to be small. So intuitively, we would expect power series like
a
0
+ a
1
x + a
2
x
2
+ a
3
x
3
+ ···
to “converge”, at least if the
a
i
are “of bounded size”. In general, the
a
i
are
“not too big” if
a
i
x
i
is genuinely a member of
x
i
R
, as opposed to some silly thing
like x
i
.
As in the case of analysis, we would like to think of these infinite series as a
sequence of partial sums
(a
0
, a
0
+ a
1
x, a
0
+ a
1
x + a
2
x
2
, ···)
Now if we denote the limit as
L
, then we can think of this sequence alternatively
as
(L mod I, L mod I
2
, L mod I
3
, ···).
The key property of this sequence is that if we take
L mod I
k
and reduce it mod
I
k1
, then we obtain L mod I
k1
.
In general, suppose we have a sequence
(b
n
R/I
n
)
n=1
.
such that
b
n
mod I
n1
=
b
n1
. Then we want to say that the ring is
I
complete if every sequence satisfying this property is actually of the form
(L mod I, L mod I
2
, L mod I
3
, ···)
for some
L
. Alternatively, we can take the
I
-adic completion to be the collection
of all such sequences, and then a space is
I
-adically complete it is isomorphic to
To do this, we need to build up some technical machinery. The kind of
sequences we’ve just mentioned is a special case of an inverse limit.
Definition
(Inverse/projective limit)
.
Let
R
1
, R
2
, , ···
be topological rings, with
continuous homomorphisms f
n
: R
n+1
R
n
.
R
1
R
2
R
3
R
4
···
f
1
f
2
f
3
The inverse limit or projective limit of the R
i
is the ring
lim
R
n
=
(
(x
n
)
Y
n
R
n
: f
n
(x
n+1
) = x
n
)
,
with coordinate-wise addition and multiplication, together with the subspace
topology coming from the product topology of
Q
R
n
. This topology is known as
the inverse limit topology.
Proposition. The inverse limit topology is a ring topology.
Proof sketch. We can fit the addition and multiplication maps into diagrams
lim
R
n
× lim
R
n
lim
R
n
Q
R
n
×
Q
R
n
Q
R
n
By the definition of the subspace topology, it suffices to show that the cor-
responding maps on
Q
R
n
are continuous. By the universal property of the
product, it suffices to show that the projects
Q
R
n
×
Q
R
n
R
m
is continuous
for all
m
. But this map can alternatively be obtained by first projecting to
R
m
,
then doing multiplication in
R
m
, and projection is continuous. So the result
follows.
It is easy to see the following universal property of the inverse limit topology:
Proposition.
Giving a continuous ring homomorphism
g
:
S lim
R
n
is the
same as giving a continuous ring homomorphism
g
n
:
S R
n
for each
n
, such
that each of the following diagram commutes:
S R
n
R
n1
g
n
g
n1
f
n1
Definition
(
I
.
Let
R
be a ring and
I
be an ideal. The
I
completion of R is the topological ring
lim
R/I
n
,
where
R/I
n
has the discrete topology, and
R/I
n+1
R/I
n
is the quotient map.
There is an evident map
ν : R lim
R/I
n
r 7→ (r mod I
n
)
.
This map is a continuous ring homomorphism if
R
is given the
I
Definition
(
I