愛知工業大学研究報告 第34号A 平成11年 l
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正則素イデアノレの記号的ぺきについてのー注意
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ARAKI
荒 木 淳
Abstract. Let k be a field of乱rbi七r乱rych乱racterisもicand let R be a k-乱1gebraof finite type. In this paper we shall show七hatforp E Reg Spec R suppose the residue class.
n
eld K of Rp is separable extension of k，
七henn
n
(p)=
p(叶 1)for alln三1. 1.Preliminaries. Throughout this paperうletus denote by k a field of arbitrary charact巴risticand letR be a k-alg巴bra. By a k-higher derivation 5 ={
5
q} of fini七erankn onR
，
we shall mean a五m七e S日quenc巴of巴ndomorphisms50，
51ぅ・・・，
5nofR as a k-vector space which satisfy七hefollowing two proper七ies: (1) 50 is the identity map ofR，
and (2) for everyr (0三
T三川，
and for allx，
yεR，
we haveι
(xy)=
L
5
i(
x
)
5
j(
y
)
i+j=r W巴shalldenoteもhecollec七ionof all suchk-higl町 deriva七ionsof五nitera山 口onR by HI~(R) On the other ha凶 le七usdenote by Derk(R) the R-module of allη-th order k-derivations ofRtoR.Thus <pεDerk(R) if and only if ψEHomk(R
，
R)，
and for allXO，
X1，
'
"
，
Xn εR w巴hav<p(叩 l'. . Xn)
=
乞
(_1)8-1L
Xi，
"
'Xi，<P(XO・hlhs
h)
8=1 I!くいくh For ev巴rycomponent 5r of 5=
{ι}εHr(R)，
5r is an r-th order derivation ofR.LetDn
denot巴the則 of compodeB6
2
5此
wh巴問 each5~? is a component of an伽 n巴叫 of Hr(R)，
and αl十 十αq三
n，
q arbitrary. For an ideal 10f R，
d巴h
Dn
(
I
)
= {]ε1 : <p(f)ε1 for巴very<pε Dれ
}
Lemma 1 ([1] Proposition 1).D
η(
1
)
is an ideal0]R
，
andωe have1"+1 CD
n
(
1
)
Lemma 2 ([1] Proposition 2). 1]Q
isαpnmαryideal 0]凡
th巴ηsoisDn(Q)
Con副 ernow a localizati叩 入 :R→
S-l R ofR.For ev町 ideal10f R letS(1
)
=
が (S-11) be七heS-saturaもionof 1.On the 0七herha吋， every high巴rderivation 5 = {ι} on R can be 巴3巾 凶 巴du阿 川Yto (1'= {5
r} on S-lR.Then letlJn d巴notethe set of composites5
2
5
p
う where巴ach(I'~?
is a compon巴ntof a unique ext巴 悶on七oS-l R of an element inHr(R)，
a~d
α1+・・ 十αqi
ミ
n.Itis clear that we have2 愛知工業大学研究報告，第 34号 A，平成 11年， Vo1.34・A，Mar.1999 where
D
e
r
k
'
(
S
-
l
R
)
is the set of alln
-
th order k-deriv.剖ionsofS
-
l
R
toS
-
l
R
.
For an ideal I ofS
-
l
R
，
denote by Ir(I)七hese七off
ε
I suchもhaも伊(1)
ε
Ifor every伊ε
Ir. Lemma3
(
[
l
J
Proposition3
)
.
I
J
n
(
S
-
l
I
)
=S
-
l
D
n
(
S
(
I
)
)
.
I
n
p
a
r
t
i
c
1
山r
，
I
J
n
(
S
-
l
Q
)
=
S
-
l
D
n
(
Q
)
f
o
r
αp
r
i
r
n
a
r
y
i
d
e
a
l
Q
o
f
R
s
u
c
h
t
h
a
t
Q
n
S
=
仇t
h
ee
r
n
p
t
y
s
e
t
.
Lemma4
(
[
l
J
Proposition4
)
.
L
e
t
入:R
→
S-lRb
e
a
l
o
c
仰t
i
o
no
f
R
，
a
n
d
l
e
t
Q
b
e
αp
r
i
r
n
a
r
y
i
d
e
a
l
o
f
R
s
u
c
h
t
h
a
t
Q
n
S
=
φ
T
h
e
n
Dn(Q)
=
入一1I
J
n
(
S
-
l
Q
)
.
2. Results. Proposition1
5
.
L
e
t
R b
e
a
k
-
a
l
g
e
b
r
a
o
f
f
i
n
i
t
e
l
y
g
e
n
e
r
αt
e
d
t
y
p
e
w
h
i
c
h
i
s
a
r
e
g
u
l
a
r
l
o
c
αJr
i
n
g
ωi
t
h
t
h
e
r
n
a
x
i
r
n
a
l
i
d
e
α1r
n
.
L
e
t
K b
e
t
h
e
r
e
s
i
d
u
e
c
l
a
s
s
f
i
e
l
d
o
f
R
.
A
s
s
u
r
n
巴t
h
αtK i
s
αs
e
p
a
r
a
b
l
e
e
x
t
e
n
s
i
o
n
o
f
k
.
T
h
e
n
Dn(m)
=m
n
+
1
f
o
r
αl
l
nミ1.
P
r
o
o
f
By Lemma 1 we ha刊mn
+1CD
n
(
m
)
.
We shall show the converse inclusion relation. Let{
Z
l
"
.
.
，
Z
r
}
be a regular syst巴m of parameters forR
.
ConsiderR
，
the貯 adic completion ofR
.
ThenR
is expressed邸 aformal power series ringK
[
Z
l
'
・
・
・
，
Z
r
]
.
Leもo
(i)={O?)h~n ε H;(長)
，
l<i<r
be七hehigher deriva七ionde:
f
i
ned byザ
)
(
Z
l
m1 • • •Z
i
mi • • •z
;
.
n
r
)
=(
7
)
Z
l
mlポー
jP3
where we put('~i)
= 0 for ]>
mi. W拙 仇=
(
Z
l
，
み )I
l
we have:I
f
]ε 仇 andifdjq)(f)ξ
的 forall九 .
，]r such出 t]1+
+
"
'
]，.~n
，
then ] E 仇n+l For， letf
=F
n
+
g
，
whereF
n
ζK
[
Z
l
'
，
・
Z，
J.is a polynomial of degreen
and 9ε
仇n
+1 Then it is 巴出ilyseen七haも 凡 =O. Assume七hatwe have already exhibited a(i) = {aJi)h三n
ε
H
:
r
(
R
)
such t11M4i)=
ザ
)
IR for all i，
j
.
Then we obtain what we want : Letf
E m be such that ψ(1)
ε
r
n
for everycpε
D
n
.
In particular，
we have。
j
J
)
3
j
J
)
(
f
)
ε
r
n
for all j1，
'
"
，
]r wi七h]l+
・
・
・ +
Jr三
η，
hence5
2
)
5
i
:
)
げ)モ仇
fo主a
l
l ]
1
，
"
'
，
jr withj1+・
・
・
+
]r~ n.This impliesF
n
= 0 and吐lUS]=g
ξ仇n
+1n
R
=
m
n
+
1
I
t
remains to show出 品 出ereexista
(
i). ={
a
?
)
}
ε
H
k
n
(
R
)
，
1三
i~ r
，
such thatザ
)
_
ザ
)IR for every i，
j.Leto'k
(
R
)
be the universal algebra of higher differ巴ntialsonR
overk
and le七o={
台}:
R
→
o'k
(
R
)
A noもeon symbolic pow巴rsof regular prim巴ideals
1コ巴thecano山 alk-higl町 deriva七ionof in五niterank (C
f
.
[
2
]
)
.
SinceK is a s巴parableextentionof
k
，
we can chooseU1γ・.'U8εR
such七hattheir images inK
form a separating transcend巴ncebase of K ov巴rk. Then Dk(R)is a fr巴eR-algebra with a free base
{6j(zz)
，
6j(um) : 1 = 1γ"，
r，
m=l，
"
・
，
s，
j= 1う
え
∞}
(
[
2
]
Theorem 3). On the0七h巴rha凶 itis easily shown七ha
:
teach6(i)= {6j勺
j，C<n εHI{(長
)
can be ir巾edd巴dinto a higher derivation
{
6
;
i)}of infi凶 erank. Hence tl日
reare un叩 巴lydetermin巴dk-high巴rd巴 町 叫ionsa(i)
ニ{
ay)}~n
R of infinite rank such七hat for all川 Jい】川m.Con悶1 required on巳s.ザ
(zz)ニ
ザ)
(zz)，
ザ
(
U
m)= 0 Theorem 6.Let k be a field of arbitrary characten:sticαnd let R be a k-algebra of finite type. For pεReg Spec R suppose the r巴sidueclαss field K ofR
v
is sepαrable extensioηof k，
then Dn(ρ
)
=
p(n十1)forα11η三1.
ProofLet入 R -
→
Rpbe七hecanonical homomorphism and set m = pRp. Then byLemma 4
，
we haveDn(p)
=
入1[Jn(m)Let{6j}だnbe a k-higher d巴rivationofRpof rank n.Then ther巴巴xistelem巴 山 tiεR
-P3 2=13 ? η
，
such that{
6
0
，
t161，
'
•
"
tnηω
6九
n}is a kι-1恒1討igh悶erd巴m乱抗七iぬonof叩2
)
.
L巴七usset。
i= ti6i，
i = 0，
1，
.
• •
，
爪
to= 1D
巴notingby {aihSnthe unique extension of{川区
πtoRp，
we have6
i = (1/ti)
a
i on Rp，
i = 0
，
1，
"
'
，
η司 L巴七伊 二 信 )
5
2
be a composite of componen凶ofhigher deri，叫ionson Rp.Then th巴reexist elem巴ntstiεR-p
，
i = 1
，
"
'
)
q，
and a family of high巴rderivations{
a
;
i)}，
i = 1・
，
，
q，
onR
such thatψ =
何)伊)
Here we denote by
a
i
i
，l七heunique ex七ensionofa
i
i
，ltoR
p
•
I
t
is obvious七hatψis阻 Rp-lin間combination of eleme山 ofDn and conseque凶 yDn(m)=げ + 1by Lemma 5. Therefore we
have
Dn(ρ)=入l(mn
+
l
)
=ρ(叫 1) for allnとlREFERENCES
1. Y. ISHIBASHI
，
Symbolic poωers of regular primes，
Can.J.Math.，
33 (1981)，
1331-13374 愛知工業大学研究報告，第34号A，平成11年，Vo.134・A，Mar.1999
2. W.C.BROWN
，
An α~pplicαtion 01 the algebra 0/ differentiαls0/ infinite rank，
Proc.Amer M叫h.Soc.35(1972)，
9-153. W.C.BROWN and W.E.KuAN
