On the dimension of discrete valuations of k ((X1, ..., Xn))
Abstract
Let v be a rank-one discrete valuation of the field Kn=k((X1,…,Xn)). We know, after [1], that if n=2 then the dimension of v is 1 and if v is the usual order function over k((X1,…,Xn)) its dimension is n−1. In this paper we prove that, in the general case, the dimension of a rank-one discrete valuation can be any number between 1 and n−1.
Full text
ON THE DIMENSION OF DISCRETE
VALUATIONS OF
k
((
X
1
;:::;X
n
))
Miguel
Angel Olalla Aos a
Faul ad de Ma ema ias. Ap do. 1160. E-41080 SEVILLA (SPAIN)
olallaalgeb a.us.es
No emb e 15, 2000
Abs a
Le
b e a ank-one dis e e alua ion o he eld
k
((
X
1
;:::;X
n
)).
We know, a e [1℄, ha i
n
= 2 hen he dimension o
is 1 and i
is
he usual o de un ion o e
k
((
X
1
;:::;X
n
)) i s dimension is
n
1. In
his pap e we p o e ha , in he gene al ase, he dimension o a ank-one
dis e e alua ion an be any numbe b e ween 1 and
n
1.
1 TERMINOLOGY AND PRELIMINARIES
Le
k
b e an algeb aially losed eld o ha a e is i 0,
R
n
=
k
[[
X
1
;:::;X
n
℄℄,
M
n
= (
X
1
;:::;X
n
) he maximal ideal and
K
n
=
k
((
X
1
;:::;X
n
)) he quo ien
eld. Le
b e a ank-one dis e e alua ion o
K
n
j
k
,
R
i s alua ion ing,
m
i s maximal ideal and
i s esidual eld o
. The en e o
in
R
n
is
m
R
n
. Th oughou his pap e dis e e alua ion o
K
n
j
k
" will mean ank-
one dis e e alua ion o
K
n
j
k
whose en e in
R
n
b e he maximal ideal
M
n
".
The dimension o
is he ansendene deg ee o
o e
k
. We shall supp ose
ha he g oup o
is
Z
.
Le
b
K
n
b e he omple ion o
K
n
wi h espe o
(see [2℄),
b
he ex ension
o
o
b
K
n
,
R
b
,
m
b
and
b
he ing, maximal ideal and he esidual eld o
b
,
esp e i ely.
Pa ially supp o ed by Jun a de Andalua, Ayuda a g up os FQM 218.
2 THE DIMENSION OF
Fix a numb e
m
b e ween 1 and
n
1. We a e ea u ing a ons u i e me ho d
in o de o ob ain examples o alua ions wi h dimension
m
.
Le us onside he ollowing homomo phism:
'
:
k
[[
X
1
;:::;X
n
℄℄
!
k
(
u
)[[
℄℄
X
1
7!
X
2
7!
u
X
i
7!
P
j
1
u
1
=p
j
i
j
whe e
k
(
u
) s ands o he algeb ai losu e o
k
(
u
) and 2
< p
3
< : : : < p
n
a e
p ime numb e s.
Lemma 1
The homomo phism
'
is one o one.
P oo :
Le us ake he elds
K
2
=
k
(
u
) and
K
i
=
k
(
u;
u
1
=p
j
3
; j
1
g
;:::;
u
1
=p
j
i
; j
1
g
)
o all
i
3.
Le us supp ose ha
'
is no one o one, hen ke (
'
)
6
=
0
g
. So le
b e
a non-ze o elemen o
M
= (
X
1
;:::;X
n
) suh ha
2
ke (
'
). Le
m
b e he
highe index suh ha
2
k
[[
X
1
;:::;X
m
℄℄.
I
m
= 1 o 2, i ially we ha e a on adi ion.
I
m
= 3, le us ake
=
(
'
(
X
1
)
; '
(
X
2
)
; X
3
)
2
K
2
[[
; X
3
℄℄
;
and onside he homomo phism
:
K
2
[[
; X
3
℄℄
!
k
(
u
)[[
℄℄
7!
X
3
7!
P
j
1
u
1
=p
j
3
j
:
We know ha
2
ke (
) and his ke nel is a p ime ideal b eause
is an
homomo phism be ween in eg al domains. We an w i e
=
g
, wi h
0
and
do esn' di ide o
g
. This o es
g
o ha e some non- i ial e ms in
X
3
. Le
s >
0 b e he minimum suh ha
X
s
3
is one o hese e ms. By he Weie s ass
p epa a ion heo em we ha e
g
=
U g
0
, whe e
U
(
; X
3
) is a uni and
g
0
=
X
s
3
+
a
1
(
)
X
s
1
3
+
:::
+
a
s
(
)
:
Sine
U
is a uni ,
g
0
2
ke (
) and
(
g
0
) =
g
0
0
;
X
j
1
u
1
=p
j
3
j
1
A
= 0
:
2
This leads o a on adi ion b eause he o o s o
g
0
a e in
K
2
[[
1
=q
℄℄, wi h
q
2
Z
,
by he Puiseux heo em.
I
m >
3 le us ake
=
(
'
(
X
1
)
;:::;'
(
X
m
1
)
; X
m
)
2
K
m
1
[[
; X
m
℄℄
and onside he homomo phism
:
K
m
1
[[
; X
m
℄℄
!
k
(
u
)[[
℄℄
7!
X
m
7!
P
j
1
u
1
=p
j
m
j
:
As in he p e ious ase we an w i e
=
h
, whe e
h
2
ke (
). So we ha e
h
=
U h
0
, whe e
U
(
; X
m
) is a uni and
h
0
=
X
m
+
b
1
(
)
X
1
m
+
:::
+
b
(
)
2
ke (
)
;
so
h
0
0
;
X
j
1
u
1
=p
j
m
j
1
A
= 0
:
Bu his is again a on adi ion by he Puiseux heo em: sine ke (
) is a p ime
ideal, we an supp ose ha
h
0
is an i eduible elemen o he ing
K
m
1
[[
℄℄[
X
m
℄.
In his si ua ion he Puiseux heo em says ha o ob ain he o eÆien s o a
o o o
h
0
= 0, like a Puiseux se ies in
wi h o eÆien s in
k
(
u
), we ha e o
esol e
a ni e numbe
o algeb ai equa ions o deg ee g ea e han 1 in
K
m
1
.
Inside
K
m
1
we an no ob ain
u
1
=p
m
and, wi h a ni e numb e o algeb ai
equa ions, we an ob ain
a ni e numbe
o p owe s o
u
1
=p
j
m
bu no all. So his
p o es he lemma.
We shall ex end o he quo ien elds his inje i e homomo phism o gi ing
an example o a ank-one dis e e alua ion o
k
((
X
1
;:::;X
n
)) o dimension 1.
Lemma 2
The e exis s a ank-one dis e e alua ion o
k
((
X
1
;:::;X
n
))
o di-
mension 1.
P oo :
We know ha he homomo phism
'
:
k
[[
X
1
;:::;X
n
℄℄
!
k
(
u
)[[
℄℄
p e iously dened is one o one. So we an ake he alua ion
=
Æ
'
, whe e
is he usual o de un ion o e
K
(
u
)((
)) in
and
'
is he na u al ex ension
o he quo ien elds. Le
b e he esidue
X
2
=X
1
+
m
2
. Hene, o ob ain
he lemma we ha e o p o e ha
=
2
k
and
is an algeb ai ex ension o
k
(
).
Le us supp ose ha
2
k
. Then he e mus exis
a
2
k
suh ha
X
2
=X
1
+
m
=
a
+
m
, so
X
2
aX
1
X
1
2
m
:
3
This means ha
(
X
2
aX
1
)
>
1. On he o he side we ha e
'
(
X
2
aX
1
) = (
u
a
)
;
so
(
X
2
aX
1
) = 1 and we ha e a on adi ion. Hene
=
2
k
.
Le us p o e ha
is an algeb ai ex ension o
k
(
). We an onside eah
elemen o
k
[[
X
1
;:::;X
n
℄℄ like a sum o o ms wi h esp e o he usual deg ee.
I
is a o m o deg ee
, hen
'
(
) =
P
, wi h
P
a p olynomial in
u
and a
ni e numb e o elemen s
u
1
=p
i
.
Le us ake
; g
2
k
[[
X
1
;:::;X
n
℄℄ suh ha
g
6
= 0 and
(
=g
) = 0. Then
'
(
=g
) =
h
0
+
h
1
, whe e
h
0
is a a ional a ion in
u
and a ni e numbe o
elemen s
u
1
=p
i
. So
h
0
is algeb ai o e
k
(
u
). Le us onside
P
(
u; Z
) =
0
(
u
)
Z
m
+
1
(
u
)
Z
m
1
+
:::
+
m
1
(
u
)
Z
+
m
(
u
)
2
k
[
u
℄[
Z
℄
a p olynomial sa ised by
h
0
, whe e
i
(
u
)
2
k
[
u
℄ o all
i
and
0
6
= 0. Le
b e
he elemen
=
P
X
2
X
1
;
g
=
0
X
2
X
1
g
m
+
:::
+
m
X
2
X
1
:
Then we ha e
'
(
) =
0
(
u
)(
h
0
+
h
1
)
m
+
:::
+
m
(
u
)
;
so
(
) =
Æ
'
(
)
>
0 and
2
m
. Subsequen ly,
0 +
m
=
+
m
=
P
;
g
+
m
:
This p o es ha
=g
+
m
is an algeb ai elemen o e
k
(
) and, a o io i, he
lemma.
Lemma 3
The dimension o a ank-one dis e e alua ion o
k
((
X
1
;:::;X
n
))
is be ween 1 and
n
1
.
P oo :
We know, a e [1℄, ha he dimension o a ank-one dis e e alua ion
o
k
((
X
1
;:::;X
n
)) is mino o equal han
n
1. So we ha e o p o e ha he e
exis s a ansenden al esidue in
.
Le us supp ose ha
(
X
i
) =
n
i
o all
i
= 1
;:::;n
. Then he alue o
X
n
1
2
=X
n
2
1
is ze o, so 0
6
= (
X
n
1
2
=X
n
2
1
) +
m
2
. I his esidue lies in
k
hen
he e exis s
a
21
2
k
suh ha
X
n
1
2
X
n
2
1
+
m
=
a
21
+
m
:
This implies
X
n
1
2
X
n
2
1
a
21
=
X
n
1
2
a
21
X
n
2
1
X
n
2
1
2
m
;
4
and hen
X
n
1
2
a
21
X
n
2
1
X
n
2
1
>
0
:
So we ha e
(
X
n
1
2
a
21
X
n
2
1
) =
m
1
> n
1
n
2
. Then
(
X
n
1
2
a
21
X
n
2
1
)
n
1
X
m
1
1
= 0
:
I he esidue o his elemen lies o o in
k
, hen he e mus exis
a
22
2
k
suh ha
((
X
n
1
2
a
21
X
n
2
1
)
n
1
a
22
X
m
1
1
) =
m
2
> n
1
m
1
. We an epea his op e a ion.
The p e ious p oedu e is ni e: i i didn' s op we would ons u he
p owe se ies
X
n
1
2
1
X
i
=1
b
2
i
X
i
1
suh ha he sequene o pa ial sums has in easing alues. Sine
K
n
is a
omple e eld, hen his se ies amoun s o ze o in on adi ion wi h
X
1
and
X
2
b eing o mally independen . So he p o edu e mus s op and he e exis s a
ansenden al elemen o e
k
in
.
Theo em 4
Le
m
be a xed numbe be ween 1 and
n
1
, hen he e exis s a
ank-one dis e e alua ion o
k
((
X
1
;:::;X
n
))
o dimension
m
.
P oo :
Le us onside he one o one ( he p oo o inje i i y pa allels ha
o lemma 1) homomo phism
' k
[[
X
1
;:::;X
n
℄℄
!
k
(
u
)[[
1
;:::;
m
℄℄
X
1
7!
1
X
2
7!
u
1
X
i
7!
(
i
i
i
m
+ 1
P
j
1
u
1
=p
j
i
j
1
i
i > m
+ 1
:
We an ake he alua ion
:=
Æ
'
, wi h
he usual o de un ion in
k
(
u
)[[
1
;:::;
m
℄℄ and
'
he na u al ex ension o he quo ien elds. We know
(lemma 2) ha he esidue
X
2
=X
1
+
m
is ansenden al o e
k
. T i ially he
esidue
X
i
=X
1
+
m
o all
i
= 3
;:::;m
+ 1 a e ansenden al o e
k
(
X
2
=X
1
+
m
;:::;X
i
1
+
m
) b eause
i
a e o mally indep enden a iables. Any elemen
=g
+
m
2
is algeb ai o e
k
(
X
2
=X
1
+
m
;:::;X
m
+1
+
m
) pa allels ha
o lemma 2. So he dimension o
is
m
.
Re e enes
[1℄ E. B iales,
Cons u i e heo y o alua ions.
, Comm. Algeb a
17
(1989),
no. 5, 1161{1177.
[2℄ J. P. Se e,
Co ps loaux.
, He mann, Pa is, 1968.
5