We define two important properties of valuations, both of which
apply to equivalence classes of valuations (i.e., the property
holds for
if and only if it holds for a valuation
equivalent to
).
Definition 15.2.1 (Discrete)
A valuation
![$ \left\vert \cdot \right\vert{}$](img1403.png)
is
if there is a
![$ \delta>0$](img1471.png)
such that for any
Thus the absolute values are bounded away from
![$ 1$](img147.png)
.
To say that
is discrete is the same as saying
that the set
forms a discrete subgroup of the reals under addition (because
the elements of the group
are bounded away from 0).
Proof.
Since
![$ G$](img31.png)
is discrete there is a positive
![$ m\in G$](img1474.png)
such that for any positive
![$ x\in G$](img1475.png)
we have
![$ m\leq x$](img1476.png)
.
Suppose
![$ x\in G$](img1475.png)
is an arbitrary positive element.
By subtracting off integer multiples of
![$ m$](img55.png)
, we
find that there is a unique
![$ n$](img19.png)
such that
Since
![$ x-nm\in G$](img1478.png)
and
![$ 0<x-nm<m$](img1479.png)
, it follows
that
![$ x-nm=0$](img1480.png)
, so
![$ x$](img178.png)
is a multiple of
![$ m$](img55.png)
.
By Proposition 15.2.2, the set
of
for nonzero
is free on one generator, so there
is a
such that
, for
,
runs precisely through the set
(Note: we can replace
by
to see that we
can assume that
).
Definition 15.2.3 (Order)
If
![$ \left\vert a\right\vert = c^m$](img1486.png)
, we call
![$ m=\ord (a)$](img1487.png)
the
of
![$ a$](img163.png)
.
Axiom (2) of valuations
translates into
Definition 15.2.4 (Non-archimedean)
A valuation
![$ \left\vert \cdot \right\vert{}$](img1403.png)
is
if we can take
![$ C=1$](img1430.png)
in Axiom (3), i.e., if
![$\displaystyle \vert a + b\vert \leq \max\bigl\{\vert a\vert,\vert b\vert\bigr\}.$](img1489.png) |
(15.3) |
If
![$ \left\vert \cdot \right\vert{}$](img1403.png)
is not non-archimedean then
it is
.
Note that if we can take
for
then we can take
for any valuation equivalent to
.
To see that (15.2.1) is equivalent to Axiom (3) with
, suppose
. Then
, so
Axiom (3) asserts that
, which implies
that
, and conversely.
We note at once the following consequence:
Lemma 15.2.5
Suppose
is a non-archimedean valuation.
If
with
, then
![$ \vert a+b\vert=\vert a\vert.
$](img1496.png)
Proof.
Note that
![$ \vert a+b\vert\leq \max\{\vert a\vert,\vert b\vert\} = \vert a\vert$](img1497.png)
, which
is true even if
![$ \vert b\vert=\vert a\vert$](img1498.png)
. Also,
where for the last equality we have used that
![$ \vert b\vert<\vert a\vert$](img1495.png)
(if
![$ \max\{\vert a+b\vert,\vert b\vert\} = \vert b\vert$](img1500.png)
, then
![$ \vert a\vert\leq \vert b\vert$](img1501.png)
,
a contradiction).
Definition 15.2.6 (Ring of Integers)
Suppose
![$ \left\vert \cdot \right\vert$](img1502.png)
is a non-archimedean absolute
value on a field
![$ K$](img9.png)
. Then
is a ring called the
of
![$ K$](img9.png)
with respect to
![$ \left\vert \cdot \right\vert{}$](img1403.png)
.
Lemma 15.2.7
Two non-archimedean valuations
and
are equivalent if and only if they
give the same
.
We will prove this modulo the claim (to
be proved later in Section 16.1) that
valuations are equivalent if (and only if) they induce the
same topology.
Proof.
Suppose suppose
![$ \left\vert \cdot \right\vert{}_1$](img1421.png)
is equivalent to
![$ \left\vert \cdot \right\vert{}_2$](img1422.png)
, so
![$ \left\vert \cdot \right\vert{}_1 = \left\vert \cdot \right\vert{}_2^c$](img1504.png)
,
for some
![$ c>0$](img1423.png)
. Then
![$ \left\vert c\right\vert _1 \leq 1$](img1505.png)
if and only if
![$ \left\vert c\right\vert _2^c \leq 1$](img1506.png)
, i.e., if
![$ \left\vert c\right\vert _2 \leq 1^{1/c}=1$](img1507.png)
.
Thus
![$ \O _1 = \O _2$](img1508.png)
.
Conversely, suppose
.
Then
if and only if
and
, so
![$\displaystyle \vert a\vert _1<\vert b\vert _1 \iff \vert a\vert _2 < \vert b\vert _2.$](img1512.png) |
(15.4) |
The topology induced by
![$ \vert _1$](img1514.png)
has as basis
of open neighborhoods the set of open balls
for
![$ r>0$](img78.png)
, and likewise for
![$ \vert _2$](img1516.png)
. Since
the absolute values
![$ \vert b\vert _1$](img1517.png)
get arbitrarily close
to
0, the set
![$ \mathcal{U}$](img1518.png)
of open balls
![$ B_1(z,\vert b\vert _1)$](img1519.png)
also
forms a basis of the topology induced
by
![$ \vert _1$](img1514.png)
(and similarly for
![$ \vert _2$](img1516.png)
).
By (
15.2.2) we have
so the two topologies both have
![$ \mathcal{U}$](img1518.png)
as
a basis, hence are equal. That equal topologies
imply equivalence of the corresponding valuations
will be proved in Section
16.1.
The set of
with
forms an ideal
in
. The
ideal
is maximal, since if
and
then
, so
, hence
, so
is a unit.
Lemma 15.2.8
A non-archimedean valuation
is
discrete if and only if
is a principal ideal.
Proof.
First suppose that
![$ \left\vert \cdot \right\vert{}$](img1403.png)
is discrete.
Choose
![$ \pi \in \mathfrak{p}$](img1527.png)
with
![$ \vert\pi\vert$](img1528.png)
maximal, which
we can do since
so the discrete set
![$ S$](img121.png)
is bounded above.
Suppose
![$ a\in\mathfrak{p}$](img1189.png)
. Then
so
![$ a/\pi\in \O$](img1531.png)
.
Thus
Conversely, suppose
is principal. For any
we have
with
. Thus
Thus
![$ \{\vert a\vert : \vert a\vert<1\}$](img1537.png)
is bounded away from
![$ 1$](img147.png)
,
which is exactly the definition of discrete.
Example 15.2.9
For any prime
![$ p$](img4.png)
, define the
![$ p$](img4.png)
-adic valuation
![$ \left\vert \cdot \right\vert{}_p:\mathbf{Q}\to\mathbf{R}$](img1538.png)
as follows. Write a nonzero
![$ \alpha\in K$](img270.png)
as
![$ p^n\cdot \frac{a}{b}$](img1539.png)
, where
![$ \gcd(a,p)=\gcd(b,p)=1$](img1540.png)
. Then
This valuation is both discrete and non-archimedean.
The ring
![$ \O$](img543.png)
is the local ring
which has maximal ideal generated by
![$ p$](img4.png)
. Note that
![$ \ord (p^n\cdot \frac{a}{b}) = p^n.$](img1543.png)
We will using the following lemma later (e.g., in
the proof of Corollary 16.2.4 and Theorem 15.3.2).
Lemma 15.2.10
A valuation
is non-archimedean if and only if
for all
in the ring generated by
in
.
Note that we cannot identify the ring generated by
with
in general, because
might have characteristic
.
Proof.
If
![$ \left\vert \cdot \right\vert{}$](img1403.png)
is non-archimedean, then
![$ \vert 1\vert\leq 1$](img1546.png)
,
so by Axiom (3) with
![$ a=1$](img1547.png)
, we have
![$ \vert 1+1\vert\leq 1$](img1548.png)
. By
induction it follows that
![$ \vert n\vert\leq
1$](img1544.png)
.
Conversely, suppose
for all integer multiples
of
.
This condition is also true if we replace
by
any equivalent valuation, so replace
by
one with
, so that the triangle inequality holds.
Suppose
with
. Then
by the triangle inequality,
Now take
![$ n$](img19.png)
th roots of both sides to get
and take the limit as
![$ n\to \infty$](img1464.png)
to see
that
![$ \left\vert 1+a\right\vert \leq 1$](img1555.png)
. This proves that one
can take
![$ C=1$](img1430.png)
in Axiom (3), hence that
![$ \left\vert \cdot \right\vert{}$](img1403.png)
is non-archimedean.
William Stein
2004-05-06