IN CHARACTERISTIC P
QUOTIENT-SINGULARITIES
BY
PESKIN
BARBARA R.
07
Harvard University
A.B.,
(1975)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June, 1980
( Barbara R. Peskin 1980
The author hereby grants to M.I.T. permission
to reproduce and to distribute copies of this
thesis document in whole or in part.
Signature of Author.
.
.
.
.
.
.
.
.
g
.
.
*1. 1*
Department of Mathematics,
Certified by
.
.
.
.
.
.
.
.
*
.
.
.
May 2. 1oR2
.
Thesis Supervisor
Accepted by.
.
ARCHNES
MAzsAczrT7
U5smu_
TECHNW!LOGY
TF
JUN i i 19-o
LIBRARIES
Chairman,
Departmental
Committee
2
QUOTIENT-SINGULARITIES IN CHARACTERISTIC P
by
BARBARA R. PESKIN
Submitted to the Department of Mathematics on May 2, 1980
in partial fulfillment of the requirements for the degree
of Doctor of Philosophy.
ABSTRACT
In this thesis we study finite automorphism groups G
of the ring R=k[[ui,...,un]] of formal power series over
an algebraically closed field k of characteristic p.
Our
interest in this problem is based on the geometrical
question of classifying the singularities which arise as
quotients of the smooth scheme Spec R by finite group
actions. We consider mainly the case that G is cyclic of
order p.
In characteristic zero, Cartan has shown that it is
possible to choose coordinates for R so that G acts
linearly, but in characteristic p this is usually not
possible. The main purpose of this work is to develop
normal forms for these non-linear actions.
Let 7 be a generator of the cyclic group G. Although
the group action cannot, in general, be linearized, we show
that coordinates can be found so that the action of G has
a partially-linear form in which at most one higher-order
term appears for each Jordan block of a . Canonical forms
for actions in which a has a single Jordan block are
developed in the cases that R has dimension 1, 2, p-l,
and
o.
Finally, we consider ways of generalizing these basic
Z/p -forms to a wider range of group actions. In particular,
we examine the actions which produce as quotients the
rational double points in characteristics 2 and 3.
Thesis Supervisor:
Title:
Michael Artin
Professor of Mathematics
3
A CKNOWLEDGEMENTS
This thesis owes its existence to the extraordinary
patience and support of two Michaels.
Michael Peskin, my
brother, has always encouraged me in my desire to do
mathematics; my advisor, Michael Artin, has shown me how.
To both, my heartfelt thanks.
TABLE OF CONTENTS
Page
.
.
.
.
.
.
.
.
.
ACKNOWLEDGEMENTS .
.
. .
.
.
ABSTRACT
CHAPTER
CHAPTER
.
.
.
.
.
.
.
.0
.
.
0
.
.
.
.
.
.0
.
.
0
.
.
.
.
.
.
0
Preliminaries.
1.
The map
2.
The depth of
3.
The characteristic zero method .
20
The One-Dimensional Forms. .
25
II.
R-G +R
RC
. .
1.
The structure of p-cyclic extensions
2.
Examples
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
0
0
.
.
0
0
.
.
0
. .
1.
A partial linearization. .
2.
The method of non-linear slicing
3.
The case
n= p-1 .
4.
The
n=
IV.
32
.
case
2
.
.
.
. .
.
.
.
.
.
.
39
.
.
.
.0
39
46
59
. ..
...
....
.
.
72
..
Examples of Generalized Actions. .
Remarks on generalized forms .
1.
2.
The rational double points
(i)
(ii)
(iii)
.
.
.
. .
Actions in characteristic 2
Actions in characteristic 3
Actions in characteristic 5
.
.
.
.
.
.
.
.
.
.
.
0 .
85
86
89
. .
Multiple-block 2Z/p-actions
2Z/p -actions for i > 1
(i)
(ii)
.
.
25
Higher Dimensions: The Basic One-Block
Actions.
REFERENCES
.
I.
CHAPTER III.
CHAPTER
.
.
.
.
.
.
.
.
.
.
.
.
.
.
96
.
.
.
.
107
0
0
0
-
111
.
.
0
0
112
.
CHAPTER I
Preliminaries
In this thesis we develop canonical forms for certain
p-cyclic automorphisms of formal power series rings over
fields of characteristic p.
Our interest in this problem is
based on the geometrical question of classifying quotientsingularities over fields of non-zero characteristic.
Let
V
be a smooth variety over an algebraically
closed field
group
G of
k
of arbitrary characteristic.
Given a finite
k-automorphisms of V, we can consider the
quotient variety V/G, whose points correspond to orbits
of points in V.
The variety
V/G
is smooth almost
everywhere, but may acquire singularities corresponding to
points of
V
with non-trivial isotropy.
One would like to
classify all singularities which can arise in this manner.
In characteristic zero, this problem has been
extensively studied and a complete classification has been
given by Brieskorn in the case that
V
has dimension two
[4 ]. The key to the characteristic zero solution is a
result of Cartan [ 5] which states that it is possible to
choose coordinates for
V
so that
G acts linearly.
Thus
the study reduces to a classification of quotients by matrix
group actions.
This reduction has the two-fold advantage
of greatly restricting the classes of actions to be
analyzed as well as of bringing the action into a form for
which explicit calculations are particularly straightforward.
char k = p
However, if
breaks down: if
divides the order of
p
of wild actions --
is non-zero, this approach
G
the case
--
it is rarely possible to linearize
the action of the group.
Consequently, very little
progress has been made in the study of characteristic
p
quotients, except in certain restricted cases in which
G
acts linearly (see, for example, [8 1).
The purpose of this thesis is to construct normal
forms for non-linear wild group actions and use them to
study the resulting quotient-singularities.
mainly the case that
G
We consider
is a cyclic group of order p.
In this first chapter we review the basic facts
about quotient-singularities, including the relevant
invariant theory, the depths of the quotient-spaces, and
the linearization methods of characteristic zero.
begin a detailed analysis of the case that
group of order
G
We then
is a cyclic
p : Chapter II deals with the possible
forms for 2Z/p-actions in dimension one and Chapter III
studies higher-dimensional actions, concentrating on the
case that
G
is generated by an element whose linear part
consists of a single Jordan block.
Finally, in Chapter IV,
we consider ways of generalizing these
wider range of group actions.
2Z/p-forms to a
The wealth of actions
possible in non-zero characteristic is illustrated by
examining the rational double points in characteristics
two and three.
§1.
The map
Let
n
RG - R.
k-algebra of dimension
R be a noetherian normal
k .
over an algebraically closed field
group
G
of
R
k-automorphisms of
the ring of invariants of
R
we denote by
,
RG
under the group action.
V = Spec R , then the quotient variety of
to the induced
Given a finite
V
If
with respect
V/G = Spec RG
G-action is defined by
Therefore, in order to study the quotient space, we wish
to examine the structure of the invariant ring RG
The ring
RG
is a normal
is local, then
RG
is local.
Proposition (1.1):
ring.
If
R
To prove normality, let
Proof:
Fract(RG)
be any element of
r
which is integral over
n-dimensional
R
r
Then
.
satisfies
an equation of the form
rm + a rm-1
where
a. E RG .
equation exhibits
over
R
itself.
+
G
m-lr
+
am
0,
G
r E Fract(RG) c Fract(R)
Now
r
...
as an element of
Since
R
,
Fract(R)
is normal, the element
so this
integral
r
lies
RG .
R ; being invariant, it lies in
in
Thus
RG
integrally closed.
finite group [ 9].
R
of maximal ideals of
RG
and
RG
R
is a
have the same dimension
RG
Finally, the maximal ideals of
.
G
Therefore, by the theorems of Cohen-
Seidenberg, the rings
n
is finite because
RG -+ R
The mapping
R
Thus if
.
are the restriction
is local, the ring
is also.
RG - R
Geometrically, the inclusion
the covering map
the order of
7: V -*V/G , which has degree equal to
G .
If the ring
R
is regular, then
is a smooth variety and the quotient-space
wherever the map
is unramified.
7
ramification locus of
p c R
locus of the map
,
(RG ) I
P
(R )P,
The image of the
V/G
be a minimal prime of the ramification
RG - R .
Let
p' = p n RG
Rp
of the map
G
(B,)G.
is smooth
be a regular noetherian normal
R
If
singular along the image of
Proof:
V/G
V
may, however, be singular in
7
Let
Proposition (1.2):
ring and let
corresponds to
Since
ht p > 2 , then
RG
is
p
and consider the localization
RG - R
p
at
p'.
Note that
is a minimal prime of the
ramification locus, the localized map is unramified except
above
p' .
It follows from the purity of the branch locus
[15] that the ring
R ,
is singular, because the rami-
fication lies in codimension greater than one.
Now Rp
is a regular ring and so R , is regular wherever the map
is unramified.
Therefore the singular locus lies entirely
along the prime
pt .
//
Because we are interested in the local behavior of
V/G
R
near singular points, we will work mainly with
If the map
a local ring.
maximal ideal of
RG -> R
ramifies only at the
R , then the quotient-space
V/G
will
have an isolated singularity at its closed point, provided
n > 2 .
We wish to study the singularities that can be
obtained as such quotients of smooth spaces.
To do this,
we will first examine some properties of the ring
§2.
The depth of
Let
let
G
R
RG
RG
be a normal noetherian local
be a finite group of
k-algebra and
k-automorphisms of
We begin our analysis of the invariant ring
RG
R
by
reviewing some facts from invariant theory and using them
to investigate the relationship between
depth R
and
depth R .
In studying rings of invariants there are two
particularly useful invariants to consider:
Definition (2.1):
Given any
x E R, the norm of x , denoted
10
Nx , is defined to be
Nx
1
=
aG
aEG
tr x , is
and the trace of x , denoted
=
tr x
Ox
aEG
are elements of the
tr x
and
Nx
Clearly both
invariant ring and so norm and trace determine mappings from
R
to
RG
x -- > Nx
defined by
If the order of
and
tr x , respectively.
x-
is invertible in R, then in fact the
G
trace map gives a surjection of R onto its invariant
subring.
If the order
Proposition (2.2):
g
of the group
G
is
prime to the ctaracteristic of k , then the map
p =
defines an
G
R.
1G
-tr:
g
R ->
RG
RG-module homomorphism which projects R onto
G
Proof:
If
Given any
x E R
p
p(r)
lies in
=
-
1
1
x= -gx
=
x
acts as the identity on the invariant ring.
fact that
R
, then
p(x)
so
r E R, its image
p
is an
The
RGmodule homomorphism also follows
easily from the definition.
R
The characterization of
the map
p
as the image of
R
under
is fundamental in the analysis of invariant
rings in characteristic zero.
In particular, we now use it
to show that the depth of
is at least as large as
RG
depth R .
Recall that the depth of a local ring
with maximal
m is defined to be the maximal length of a regular
ideal
m , i.e. the largest integer
sequence in
is not a zero-divisor in
x
is not a zero divisor in
d
for which
xl,...,xd E m such
there exists a sequence of elements
that
R
and the image of
R
R/(x 1 ,...,xi
1
)
We will also
.
use the equivalent cohomological formulation that
is equal to the smallest integer
where
such that
If the order of
teristic of
k
then
= RG/InR.
(R/I)
and if
The image
p(I)
I
of
G-invariant ideal of
is a
I
is contained in
.
Now
p(I) c I n p(R)
# 0
We therefore have
G
.
I
Thus
R
p
,
because
o(R/I)
and the opposite
inclusion follows from the fact that
map.
H$(R)
is prime to the charac-
G
the ideal is stable under the action of
o(R)/p(I)
depth R
denotes local cohomology (see [1o]).
Hm
Lemma (2.3):
Proof:
i
x.
is a projection
=
12
(R/I) G
=
G be a group of
Proof:
Let
zero-divisor in
RG
any
Then its
s E R
(Nr)RflRG
such that
=
(Nr)s
which implies
(R/(Nr) R)G
s = ps
=G/(Nr)RG
obvious.
be an element which is
norm
By the preceding lem ma,
Now the ideal
R
k-algebra
with order
depth RG > depth R
r ER
R.
RG/I nRG
k-automorphisms of
depth R = 0 , the result is
If
depth R > 0
in
Then
char k.
prime to
=
R be a noetherian local
Let
Proposition (2.4):
and let
p (R)/ITn p(R)
=
p(R/I)
Nr
not a
is not a zero-divisor
R /(Nr)R rRG= (R/(Nr)R)G
(Nr) R
is equal to
(Nr)s
Assume
, for given
E RG , we have
(Nr)ps
p((Nr)s)
and hence
s E RG
Therefore
and so
depth(R/(Nr)R)
=
depth(R/(Nr)R)G =
depth R -
depth RG - 1
.
//
The result now follows by induction.
As an immediate consequence we have
Corollary (2.5):
Assume that the order of
the characteristic of k.
the invariant ring
RG
Then if
R
is
G is prime to
Cohen-Macaulay,
is Cohen-Macaulay.
13
R
since
is
RG -*R
because the map
have the same dimension
RG
and
R
The rings
Proof:
finite.
dimR = depthR
Also,
It follows from the proposition
is Cohen-Macaulay.
that
>
dim RG
depth RG
>
depth R
=
dim R
Therefore equality holds throughout and the ring
is
RG
//
Cohen-Macaulay.
Consequently, quotient-singularities in characteristic
zero are always Cohen-Macaulay since they are defined by thering of invariants of regular, hence Cohen-Macaulay, rings.
We now examine how this situation changes when the
order
g of the group
G is
divisible by the characteristic
of k.
If
char k = p
is non-zero then the operators norm
and trace still define mappings from R to RG, but now the
characterization of
In fact,
down.
if
RG
as the subring of traces breaks
p divides
g, then given any
x E RG
,
we have
tr x
ax
=
=
x-g
=
0
aEG
Thus all of
RG
G
lies in the kernel of
tr: R - R
G
Because of this collapsing of the trace mapping, the
G
above results are no longer valid when the order of
divisible by the characteristic.
is
The remainder of this
section is devoted to an analysis of the depth of
R
for
these wild group actions.
(Fogarty, [7 ])
Proposition (2.6):
depth R > d
If
If
Proof:
d= 0
d = 0, 1, or 2.
Let
depth RG > d .
then
the result is clear.
depth R > 1
If
,
then there is an element r which is not a zero-divisor in
R .
Its norm
Nr
and so
RG
is not a zero-divisor in
depth RG >1.
Now assume
depth RG > 2
consider the quotient map
R-*R/(Nr) .
on invariant rings which has image
d=1
case of this proposition.
element
i
has the form
invariant in
Z
(R/(Nr))
Z
.
s+h
for some
.
Therefore
In particular,
by the
h E (Nr)
depth(RG/(Nr)RG) > 1
and so
since
,
ep(Ns)= 9g , where
Zg
(R/(Nr))
Lift
a E G , the element
lies in
is not a zero-divisor in
is not a zero-divisor in
.
s
is
g
is
RG/(Nr)RG.
RG/(Nr)RG
because
Therefore
depth(RG) > 2
Unlike in characteristic zero where
with
,
Therefore there is an
Given any
s E R
R/(Nr)
the order of G.
Moreover,
The ring
Z E (R/(Nr))G which is not a zero-divisor.
to an element
as
is
ep: RG ->(R/(Nr))G
RG/(Nr)RG .
has depth > 1 and hence so does
R/(Nr)
Nr
Because
G , there is an induced map
fixed under
as above and
Nr
Choose
.
.
depth RG increases
depth R , the depth of RG is often bounded in non-zero
characteristic.
In particular, we will show that the
d > 2.
statement of Proposition (2.6) cannot be extended to
For example, if
is a cyclic
G
Let
Proposition (2.7):
noetherian local
p-group we have
char k= p and let R be a normal
Let G be a group of
k-algebra.
k-automorphisms of R, cyclic of order
Then
and assume
,
pcR.
is ramified only along the prime
RG-+R
that the map
pV
depth(RG) < 2 + dim(R/p)
Let R be a noetherian local ring and M a
Lemma (2.8):
Then
R-module.
finite
for all primes
depthM < dim (R/p)
p E Ass(M)
If
Proof:
depthM = 0,
depthM > 0 and let r
R
the result is
be an element in the maximal ideal of
which is not a zero-divisor on M .
depthM'
= depthM - 1
depthM'
< dim (R/q)
and it
Set
M'= M/(r)M.
Then
follows by induction that
for all primes
q e Ass(M')
therefore suffices to show that for any prime
there exists a prime
Assume
trivial.
such that
q E Ass(M')
.
It
p E Ass(M)
q = p+(r)R.
This will imply that
depthM - 1
since
r E q
Let
m
but
depthM'
=
r
< dim(R/q) < dim(R/p) - 1
p.
be any element of
M
for which
Ann(m)
=p.
16
Since R is noetherian local, there exists some integer
m E rNM
such that
some
x EM
Ass(M')
r N+1M.
i
The image
containing
q
of
in
m as
rNx
M'= M/(r)M
for
is non-
There exists an element of
.
Ann(x)
and this gives the desired
If R is a noetherian local ring and p is a
prime ideal of R., then
If
depth R < depth (R ) + dim (R/p)
depth R = 0 , the lemma is
depth R > 1 and that there is
zero-divisor in
and,
x
Write
.
Lemma (2.9):
Proof:
m
Ann(i) -J p +(r)R
zero and
prime
but
N
setting
R.
Then r
R' = R/(r),
some
r Ep
Assume that
which is not a
is not a zero-divisor in
we have
depthR1 = depthRp - 1 , and
clear.
R
depth R1 = depth R - 1
R/p = R'/p n R'
since
r E p.
Proceed by induction.
If p contains no
prime
lemma,
since
R-regular element, then there is a
g E Ass(R) which contains p
depth R < dim (R/q)
and
.
By the preceding
dim (R/q) < dim (R/p)
p c q
Lemma (2.10):
(Fogarty, [7 ]) Let k be a field of charac-
teristic p and let R be a noetherian local
Let G be a group of
p
k-algebra.
k-automorphisms of R, cyclic of order
, and assume that the induced action of G on
free except at the closed point.
depth RG < 2 .
If
Spec R is
depth R > 2 , then
17
Proof:
Let X denote
Spec RG
-
{m,} , where
m n RG
m' =
Spec R - {m)
.
is the maximal ideal of R and
By assumption, the map
We want to show
H I(R) /
m
and let Y denote
depth RG < 2
X-+Y
is unramified.
or, equivalently,
By the local cohomology sequence
.
H1- (RG
and the fact that
H(R)
H1(RlG
H (RG) = 0
,
H
(RG
G
it suffices to show that
=H(Y) /
{m'})
-
H2(RG-
0
Consider the spectral sequence for unramified covers whose
term is
E2
E'
(2.11)
HP(G,H (X,X))
We will show that the
term injects into
Therefore the
if
a
E2
H*(Y,y)
E2 ,0-term is non-zero.
H1 (Y,,y)
By hypothesis,
==
Since this
this will complete the proof.
depth R > 2
and so
-term is equal to
H0 (X,OX) = R .
H (GR).
is a generator of the cyclic group
G
,
Recall that
then
H1(GR)
is defined to be
H (GR)=
(2.12)
{elements of R with trace 0}/{image of (a-l)}
Now given any
a
is a
k-automorphism.
tr(l) =p'-1
E12
0
x E R, the element
0 .
Therefore
This implies that
does not vanish.
ax- x
1
g
lies in m since
Tm(a-1), but
H (G,R) /0
and hence
18
Let_pt
Proof of Proposition (2.7):
of
p
that
to
and consider the map
RG
(RG)p,
(R,)G
=
,
dim(RG,
depth(Rp ,) > 2
(RG)p t ->Rp
Note
.
and that the localized map is
ramified only at the maximal ideal.
depth (R
denote the restriction
< 2
If
ht p' > 2
If
.
ht p' < 2
then
then
by Serre's criterion for normality.
Therefore we can apply the preceding lemma to conclude
depth(RG) , < 2 .
again that
since the map
RG ->R
Also, dim(RG /p') = dim(R/p)
is finite.
Therefore, by Lemma (2.9),
depth RG < depth (RG) , + dim (RG/p)
< 2 + dim (R/p)
Proposition (2.7) implies that the bound on the
depth of
RG
is determined by the dimension of the
ramification locus.
In particular, for the case of isolated
quotient-singularities we have
Let
Corollary (2.13):
be a regular noetherian local
R
k-algebra of dimension > 2
ideal of
Proof:
.
If
G
R , cyclic of order
k-automorphisms of
p = char k
and let
RG ->R
R , then
be a group of
p , where
ramifies only at the maximal
depth (RG)
-
2 .
By Proposition (2.6), depth RG > 2
hand, Proposition (2.7) implies that
.
On the other
depth RG < 2
.
/
19
Thus, although quotient-singularities in characteristic zero are always Cohen-Macaulay, isolated singularities
arising from wild group actions are Cohen-Macaulay only in
dimensions < 2.
Finally, we remark that Ellingsrud and Skjelbred have
shown the inequality of Proposition (2.7) can be strengthened
to equality in the case that
the action of
G
is a polynomial ring and
is linear.
Proposition (2.14):
R
R
(Ellingsrud and Skjelbred, [6 ])
Let
be a polynomial ring over a field of characteristic
and let
G
be a group of linear automorphisms of
order
p
prime
p C R , then
.
If the map
RG - R
R
p
of
ramifies only along the
depth(RG) = min(2 + dim(R/p), dim R)
We refer the reader to [6 ] for the proof.
Note that
Proposition (2.14) also holds for a formal power series
ring
R
,
since depth is preserved under passage to the
completion.
20
$3.
The characteristic zero method.
In this section we review the methods used to classify
quotient-singularities in characteristic zero.
Because
we are interested only in the local behavior of the
quotient space near its singular points, we will work with
complete local rings.
Therefore let
R = k[[u ,...,un
be the ring of formal power series in
algebraically closed field
group of
and let
k
k-automorphisms of
R
.
letters over an
n
be a finite
G
In order to classify
Spec RG
the possible quotient-singularities arising as
we wish to develop normal forms for the action of
Let
a E G
R .
be an automorphism of
can be realized as a system of
n
of
a
in
u ,...,un , one for the image of each
Such a system defines an automorphism of
G
on
R
The action
power series
under
u
R
a
provided the
linear terms of the power series define an invertible
linear transformation.
au
=
a
au
=
anlu
Therefore
u1 + a1
2 u2
+
...
a
has the form
+ alnun + h (u,...,un)
(3.1)
where
+ an 2 u2 + ... + annu
detla.
a.. E k are such that
has order
> 2
.
Denote by
+ hn(u ,...,u n),
0 and
(a.)
E R
the invertible trans-
a
formation determined by the linear terms of
matrix action given by
h
.
a
,
i.e. the
21
Using a variant of the trace map
p
of the preceding
section, it is possible to define a coordinate system for R
so that
a
acts as a linear transformation.
[ 5 ])
(Cartan,
Proposition (3.2):
Assume that the order g
of the group G is prime to the characteristic of k and
5i
define elements
L
=
-
i =1,.9e*..,n
~Aa(ui) ,
.
aEG
form a coordinate system for R for which
(5 }
Then the
of R by
the action of G is linear.
A map
Proof:
u
"+
defines a change of coordinates
provided the linear part of the map is an invertible transformation.
For the above choice of
ui
verified, because the linear part of
this is easily
a
is
the identity
transformation on R.
Now consider the action of G in the new coordinate
system
(5 } .
T E G , the image of
Given any
Tu
in the
new system is
3~AaTu
(i~~~A(aT)ui
- I(-fr)~A(CT)ui
1-1
=
Therefore the action of
action of
on
u.
u
T on u
.
transforms to the linear
//
22
Thus if the characteristic of k is zero, the problem
of classifying quotient-singularities for all finite automorphism groups G reduces to the one of classifying
quotients under matrix group actions.
and
--
--
k=0
In the case
n =2
i.e. the case of complex surface singularities
Brieskorn has given a complete classification of quotient-
singularities by cataloguing the possible finite subgroups
of
GL(2,W)
[ 4].
and examining the corresponding quotients
We briefly sketch the results for cyclic groups.
Example (3.3):
The case of a cyclic group
with order n and generator
a
G c GL(2,C)
Since G is a group of
.
order n , the eigenvalues of a are nth roots of unity,
say
C and
('.
Therefore we may assume that
a has the
form
20
0
If the map
and
C'
some
i
2
-
2/G
ramifies only at the origin, then
are primitive roots of unity and so
relatively prime to
If
V)'
C'=
C
for
n.
C'= (- , the invariants of a are generated by
x =u,
y= u
, and
z = u u2
and the invariant ring RG is defined by k[[x,y,z]]/(znxy).
This is a rational double point of type
At the other extreme, if
generated by the
n+1
monomials
C'=
,
An-1 *
the invariants are
23
n
n-l
U1, u1
u 2 ,..,
n-l
ulu 2
n
,u2
and the resulting singularity is the cone over the twisted
n-ic, a rational singularity of multiplicity n.
The singularities resulting from the other choices
for
C'
are also rational, with multiplicities ranging
between 2 and n.
(See [4 ] for a complete listing.)
C -C2 /G
If the map
ramifies in dimension 1, then
the resulting quotient-singularity is equivalent to the
quotient by the smaller group
G'= G/H
,
where
H is the
largest subgroup of G fixing the ramification locus.
map
02 2/G'
the action of
The
will then be unramified in dimension 1 and
GI
will have the above form for some
n'
dividing n.
This example is indicative of the methodology used in
the characteristic zero classification.
For all the
quotient-singularities, one first finds an appropriate
matrix form for the action from which invariants can (in
theory) be computed.
The resulting singularities are then
classified and, in characteristic zero, all such quotientsingularities are rational.
Over fields of non-zero characteristic
p , the above
analysis is still valid provided p is prime to the order
of G.
In contrast, if
the order of G is
divisible by p
then it is rarely possible to linearize the group action.
2~4
Proposition (3.4):
R= k[[u ,...,un]]
Let
and let
G be
a group of
k-automorphisms of R with order divisible by
p= char k.
If
RG -R
ramifies only at the maximal ideal
of R, then it is not possible to choose coordinates for
R
so that
Proof:
G
acts linearly.
Assume that coordinates u
the order of G is
call it
p-divisible,
could be found.
there is
a, which has order p.
Since
an element of G,
We may assume that the
coordinates u
have been chosen so that
form.
a = 1., every eigenvalue of a is a pth root
Because
of unity:
a is in Jordan
in characteristic p , this forces every eigen-
value to be 1.
(3.5)
Therefore a has the form
au1
=
u
au2
=
u2 + e1 u1
YUn
=
un + e
where, for each i, the number
un-lunc
is either 0 or 1.
this action fixes the 1-dimensional locus
= un-l= 0} , i.e. the
assumption that
RG-R
{u = u2
But
='''
un-axis, which contradicts the
ramifies only at the maximal ideal.
//
Therefore wild group actions cannot, in general, be
brought into linear form and so it is necessary to develop
new normal forms for these non-homogeneous actions.
To
understand what forms these actions can take, we turn first
to an examination of the case that R has dimension one.
CHAPTER II
The One-Dimensional Forms
Let
k
be an algebraically closed field of
characteristic
p .
We begin our analysis of wild auto-
morphisms of formal power series rings
examining the case that
automorphism group
§1.
G
The structure of
R
R
over
k
by
has dimension one and the
is cyclic of order
p
p-cyclic extensions.
Our main result on the structure of the extension
RG ->R is the following:
Theorem (1.1):
R = k[[u]]
Let
be a formal power series
ring in one letter over an algebraically closed field
of characteristic
p .
Let
G be a
p-cyclic group of
k-automorphisms which acts non-trivially on
generated by an element
a .
R
such that the invariant ring
(ii)
where
RG
u
for
Fract(R)
is generated by an element
z
R
is equal to
x = Nu;
the field extension
relation
and is
Then
(i) there exists a choice of coordinate
kEx]],
k
over
Fract(RG
satisfying the
26
(1.2)
zp -
(iii)
z
for some integer
i > 0, relatively prime to p ;
the action of
on
a
R
has the form
au = u + h ,
(1.3)
where
h E R
has order
i+l, and this form is
independent of the choice of
Lemma (1.4).
is equal to
k[[x]]
The ring of invariants is normal and 1-dimensional,
hence is regular.
has the form
Being a complete local ring, it therefore
k[[x]],
by Cohen's Structure Theorem.
Let
Proof of Theorem (1.1):
RG = k[[x]].
so that
a
R
.
x E RG *
for some
Proof:
The invariant ring
U
x
be chosen as in the lemma
The field of fractions
p-cyclic extension of
//
Fract(R )
Fract(R)
is
and therefore, by
Artin-Schreier theory, it is generated by an element
z
satisfying the relation
zP - z
where
t
as
a
ax
E k
-iG
and
t
i >0
for some unit
to the equation
x---> cx
=
c
=
is
a Vx
relatively prime to
a E RG
If
c
p.
is a solution
= a , then the coordinate change
results in the new relation
Write
27
zP - z
x-i
=
,
and so gives the desired form (1.2), proving claim (ii).
Now the action of
a
on
az
=
z + 1
(1.5)
and therefore the norm of
Nz
=
z
Tla'z
z
is
given by
is equal to
=
-
z
x~
=
To establish claim (i), it remains to show that there exists
an element
u
generating
Choose the element
u
j
-i mod p
Let
x
as its norm.
n
be the integer
and such that
ln <p-l.
to be the smallest positive integer for which
(1.6)
in
u = xjzn
Set
which has
as follows:
which is the inverse of
Choose
R
.
=
jp-1
Then
Nu = (Nx)3 (Nz )n
= xpji -in
= x,
as required.
.
The complete local ring
R
has a natural valuation
given by the order of power series and this is independent
of the choice of coordinate.
generates
Therefore, to show that
R, it suffices to show that
ord u = 1
.
u
Now
28
ord(u) = ord(avu)
the extension
R
v , and
for all
over
RG
ord(x)
p , since
=
is Galois of rank
p and is
totally ramified at the origin. Hence
p-1
p.ord(u) =
ord(avu) = ord(Nu) = ord(x) = p
v=0
and so ord(u) = 1. This completes the proof of (i).
To establish (iii), first observe that
since
ord(u) = 1, ord(x) = p, and
of
on
a
z
u = xiz.
az = z +1
is given by
au = a(xizn) =
ord(z) =-i,
The action
and therefore
x(z+1)n
= xizn + nxizn-1 + ...
= u + h ,
where
ord(h) = pj - i(n-1) = pj -
in + i = i + 1.
Although the choice of the coordinate
canonical, the order of
u
Any other coordinate
u
for some
a5=-
a
E k
for
R
is not
is independent of choice:
has the form
a1u + a2u2 + a3u3 + .. *
with
a
#'
0 .
= a1 (h) + a 2 [ (u+h)2
= h(a
Thus
=
au- u
u
Therefore
u 2] + a3 [(u+h) 3 -u 3 ] + ..
+higher degree terms)
ord h = ord h .
.
29
We next wish to find the equation for
ring
RG.
u
over the
In order to do this, we first need to introduce
certain symmetric polynomials which arise as coefficients.
Lemma (1.7):
Let
polynomial on
denote the
sm
p
mth
letters and let
elementary symmetric
be a root of the
equation
zp -
z
ii
=
.
Then
sm
=
,1
0
if
m < p-1
-1
if
m = p-1
if
m= p
M-t+1
s-1 _- it
x~
and
p-1
qm =
for
L(
+2 )m
m < p(p-1), where
such that
Proof:
s
m = s(p-1) + t
The formula for
and
and
sm
t
are non-negative integers
0 < t < p-l
follows at once from the fact
that
0
The formula for
q
=
(z -)(z-
=
zP
-
z
-
(i+1))
x-i
...
(z- ( +p-1))
is derived from the general formula
of Girard [11], which expresses the symmetric function
Mmin terms of the elementary symmetric functions on
i=l
the
a 's
30
p
a,
= (-1)
(le -1) !m/e
where the summation is over all
ie. = m
and
e
and
sp -
(el,e 2 , ...
)
for which
=
S
.
+
,
all
the
s.f s
ep-1 (P-l)
in
(1.9)
0 < m< p(p-1)
vanish
Therefore the only non-zero
terms of the summation result from expressing
Now
.9
> 0
In the case that
except for
te>eie2
2 a0 0
!e2!... s
M
in the form
+ eP(p)
< p and 0 < e
0 < e
In this range there is at most one choice of
ep and
and so
< p-l.
eP
1
satisfying (1.9) and the summation (1.8) collapses to
p-1
1= m+e
= (-1)
Write
m
p
eP((e
in the form
m
=
+eP-1) !m/e_
P !ep !)(-1)-lxs(p-1)+ t
where
p
0 < t < p-l . Then
s(p-1) + t
= (s-t)(p-1) + t(p)
and
e _1 = s - t
and
e
= t .
The formula therefore
becomes
-it
~~mns
= -)
=(-1)m-t+1
- ((s-1)!(-1)(s-t-)/(s-t-l)!t!(s-t))xi
s-t1 )x-it
Note that if no choice of
range
0
then
p-1 I
to
and
eP
s < t
e
and so
exists in the
= 0
(S1
corresponding to the fact that the summation is 0.
Theorem (1.10):
u = x jzn
Let
//
be chosen as in Theorem (1.1).
Then the integral equation for
p-1
(1.11)
U +
La(-1)a
u
over
k[[xJ]
x
u
=
given by
is
x
,
2%=1
where
a,
(1.12)
e1 e2
+e
i(e
e2
..
/l
2
Qq
=
a~~~-l
are defined as in lemma (1.7) and the
q
The elements
summation (1.12) is
and
= A
Lie
Proof:
11
where
e
over all
(el,e 2 ,...)
such that
> 0 .
The equation for
p-1
.e le2 '''
(z- ( +A))
z
over
=
zP -
Fract(R G)
- x~
=
9 is any fixed root of the equation.
u = x z
is
given by
0
The element
therefore satisfies the equation
p-1
R
(u - x (+
)n)
=
0
Expanding this product, we obtain
p
uP +
Now
(-1) x
s (gn,.,
p
s
+ p_
(-n(
1 )n)
1n,...,$(
+p-1)n up-A
=R g+4n = s (,gl
=
0
. .9, +p-1) ]n=
32
-in
x
,
and so the last term of the summation is
(-1) p jpx-inu
=
.
-x
The remaining terms can be calculated using the formula
inverting (1.8)
[11], namely
L
sja.)
l+e.,(,
D-(1.L2
el 1,2 )e2.e
where
(a) = (ac,...,a
)
all
,e 1 'e2 !,,.)
and the summation is taken over
= I
for which lie
(e1 ,e2 ,...)
1e2
/ 1 2
Evaluating this equation for
(0)
and
e
> 0 .
(,n,.
,+pl)n)
results in the formula (1.12).
Although the
a.
may contain negative powers of x.,
the total exponent of
x
(1.11) is at least 1.
This follows from the fact that
au = u+h
x
ord(h) = i+1
where
=
Nu
in each term of the summation
=
u
, so that
+ (terms of order > p+i)
//
§2.
Examples.
The equation for
u
is, in general, far simpler
than Theorem (1.10) suggests because almost all terms of
the summation (1.12) vanish.
Consequently, it is usually
easier to derive the integral equation for
ring
RG
zp- z = x~
u
over the
by computing directly from thte equation
defining the field extension.
In this section
we illustrate the forms the resulting equations can take by
computing the possible formulae for several choices of
p
i
and
.
Consider first
some
m > 0
the case
p = 2.
i = 2m+l
Then
for
and so, with notation as in the proof of
Theorem (1.1), we have
u = xjZn = xm+l z .
n = 1
j = m+1
and
.
The equation satisfied by
Thus
u
can be
derived from the equation
Z2 -
by multiplying through by
clear denominators.
x
Z=
x-(2m+l)
x 2 (m+l)
pj
in order to
We obtain
2m+2 2 x 2m+2
z-x
z=x
or
u2 -x m+lu
(2.1)
x
=
.
This is the basic equation for all characteristic 2
forms.
The action of the automorphism
au
=
a(x
a
is given by
m+l(z+1)
z)
u
+ x
m~l
The equation (2.1) can be solved recursively for
terms of
u
terms of
u
x
in
to obtain a closed form for this action in
m+1
alone. We note that Nu= u(u+xm) _
u - x m+u , which agrees with (2.1).
If
p
=
3, there are two possible forms for degree
p
34
extensions, depending on whether
the first case,
j
=
i =
u = x
2m+l, and
3m+l for some
2mn+1 2
z
.
In
i = 1 or 2 (mod 3).
m > 0
and so
The equation for
u
n = 2,
is
obtained by squaring both sides of the equation
z3
-
z
and multiplying through by
X6rm+3 6
(2.2)
x pi
_ 2 i6m+3 4 +
U
The action of
=x-(3m+)
a
2x 2
-
on
z
u
=
6m+q
=
x
u+2 =
x
6m+3 2
u2 + x
can be computed using the fact that
=
l/x3m+ (z2-1
=
l/xm u- x2m+1
It follows that
au = a(x
2mr+1 2
z ) = x2mn+1 (z+1) 2
2m+l z2 + 2x2m+lz + x2m+1
= u + (2xm+
/(u-x2m+))
+ x2m+1
and again a closed form can be obtained by solving
equation (2.2) recursively for
If
i = 2 (mod 3), then
and the resulting form is
equation.
action for
We have
a
is
x
i = 3m+2
u
for some
m > 0,
analogous to the characteristic 2
n = 1, j = m+l,
given by
in terms of
and
u = xm+lz
.
The
35
au
(z+1)
=
xm+1
=
u + xm+1
and the resulting equation is
x
Nu
=
(2.3)
=
u(u +xm)(u+ 2xm
=
u3 - x2(m+l)u
In general, the form of the equation for
determined, up to changes in the exponents of
alone and hence depends only on the residue of
u
x , by
n
i mod p
(This can be seen in the formulae given in Theorem (1.10).)
Therefore, to simplify notation, we will deal only with
equations arising from values of
i
between 1 and
p-1
for the remainder of this section.
If
njp-1 , as in all of the above examples, the
formula for
Nu
is particularly easy to determine.
i < p-1 , this condition is equivalent to say
Proposition (2.4):
for
u
(2.5)
in = p-l
Assume
.
Since
in = p-1 .
Then the equation
is given by
( )x ui
(-1)
and the action of
a
on
u
i
=
x
has the form
n
(2.6)
( )(x/(u -X )-
au = u +
2=1
Proof:
From the equation
in = p-l,
it follows that
j= 1
36
u = xzn.
and therefore that
The relation for
u
is then
derived from the equation (1.2)
Z
Z
z~nz
-i
=
x~
defining the field extension, by raising both sides of
this equation to the
by
xpj = x
.
nth
power and multiplying through
We obtain
n
p- in
(p-
pznp-
-
1=0
n
(-l)z(n)xuzn(p-i)
=
Z=0
n
(-1) )x
Again
z
.
u
using equation (1.2), it is possible to solve for
in terms of
u:
z
1/(x i(zp
=
xi
zin
S1/(Xi
=
- 1))
1/(u -x )
Subsituting this into the formula
Cu
=
a(xzn)
=
x(z+1)n
=
u
+
(n)xzn-,e
results in the equation (2.6).
The first case in which
n jp-1
is when
p = 5
37
i = 3.
and
n = 3, j = 2
In this case
u = x 2z3 .
and therefore
u , we again raise
To find the equation for
the relation
z
z5 -
to the
nth
=x-3
power and multiply through by
x
.
This
results in the formula
x
=
x 1 0 (z
=
xz
=
u
=u
5
5
15
3z 1 1 + 3z 7 -
-
3xz (z 5 - z)
-
-
3x u2(x 3)
-
- x 3 u-2
u
x8 u
n 1 p-l
forms, the exponents
do not decrease by a constant multiple nor do the
exponents of
to
- x1 0 z'
x 8 u.
Note that here, unlike the
of
-
z3 )
x
and
u
within each term necessarily add
p .
p = 5
The remaining cases for
are all of the form
given in the proposition, and so the complete listing for
p =5
is
- 4xu 4 +6x2u3 - 4x3u2 +x
i=1
x= u
i =2
x = u 5 - 2x2 u3 +x4u
3=i=4
-5
x =u
-
4
u
3 2 - x 8u
xu
x = u 5 - x4 u
More generally, given any
to obtain the equation for
u
p
and
i, it is possible
by methods identical to
38
those used above:
is raised to the
the equation defining the field extension
nth
power, denominators are cleared
by multiplying through by
xpj , and the resulting equation
is then translated from an equation in
in
x
and
u.
x
and
z
to one
However, this last step may become quite
complicated, depending on the relationship between
i
p-i, and as a result the general rule simplifying the
formula of Theorem (1.10) has not yet been found.
and
39
CHAPTER III
Higher Dimensions:
The Basic One-Block Actions.
In this chapter we examine
p-cyclic actions for
rings of dimension greater than one.
and let
G
of order p.
be a group of
Let
R = kLu1 ,....,un
k-automorphisms of
R
,
cyclic
We wish to study the possible structures of
the invariant space
RG
by first constructing canonical
forms for the action of G.
Although the group action cannot, in general, be
linearized, we show in §1 that it can be brought into a
partially linear form.
We then turn to a detailed analysis
of the case that the linear part of the action consists of
a single Jordan block.
In §2 a method is developed for
studying the resulting invariant ring as the slice of the
invariant ring for a linear action on a higher dimensional
space.
The remainder of the chapter is devoted to applying
this method to the study of rings of dimension 2 and
§1.
p - 1.
A partial linearization.
As in the characteristic zero analysis, we would like
to choose coordinates for R so that G acts in as simple
a form as possible.
However, in the case that the map
RG -4R ramifies only at the maximal ideal
m c R ,
]
4o
Proposition (1.3.4) shows that it is not possible to
linearize the action of G.
In this section we develop
instead a partially linear form for the group action and
set up a program for analyzing these non-linear forms by
means of linear models.
a be a generator of the cyclic group G.
Let
Because the obstruction to linearizing the group action is
controlled by the ramification locus of the map
RG -+ R
we first give a criterion for determining when the map
m .
ramifies only at
Let
Proposition (1.1):
a cyclic group of
Let
R = k[[u l,...,un]]
k-automorphisms of
a be a generator of G.
and let
R
G
be
of prime order.
Then the map
RG -R
is
unramified in dimension > 0 if and only if the ideal
r = (au - u1 ,...,aun- un)
Proof:
p
The map
is fixed by
is
m-primary.
RG -+R is ramified along the prime
a
and the induced action of
a
on
p
if
R/p
is trivial.
Assume that the map ramifies along a non-maximal
prime
p
Because the action on
.
follows that
and so
r
au - u. E p
is not
Conversely,
non-maximal prime
R/p
is trivial, it
for every i
Therefore
r C p
m-primary.
if
p
r
is not
m-primary,
containing it.
there exists a
Therefore
au 1--u1 E o
41
for all
mod p
i
and so
is trivial.
p
is fixed by
Hence
RG -R
a
and the action
ramifies along
To study the ramification locus of
RG -R
p .
///
it therefore
suffices to study the primes associated to the ideal
r = (cu1-u 1 ,...,aun-un)
Now any automorphism
a
of
R
can be represented
in the form
au
=
a u 1 + ..
aun
=
anlu + ... + annun + hn(u ,...,un) '
and
hi E R
+ alnun + h (ul,...,un)
(1.2)
where
a.. E k
linear part of
has order > 2. Denote the
a , i.e. the invertible transformation
determined by the matrix
(aij) , by
' .
change of coordinates we may assume that
form.
All the eigenvalues of
since
a
has order p ;
B
are
pth
in characteristic
that all eigenvalues are 1.
By a suitable
3
is in Jordan
roots of unity
p
this means
3
The matrix representing
therefore composed of blocks of the form
1 1.
Lemma (1.3):
is
p
O
The maximal size of a Jordan block of
F
is
42
The automorphism a , and hence its linear part
Proof:
has order
p.
Therefore
is nilpotent of index
-1
',
< p
for
(a-1)P
p
=
It follows that no Jordan block of
-1
3
0
=
.
can have dimension
greater than p .
//
The action of
a
itself is therefore defined by
+
blocks of the form
au
=
U
au i+1
=
Ui+1 + u
(1.4)
au +j
where
h
E R
at most p
h
(ul,...un
+ hi+1(u
,...,nI
ui+j + ui+j-1(u,...,un
has order > 2 and-the block size = j+1
If the map
RG -R
is
is unramified in dimension
>0, it is not possible to eliminate all of the higher order
contributions
h
within each block.
However, it is
always possible to partially linearize the action:
Proposition
a
k-automorphism of
Let
R
with order p .
a choice of coordinates for
blocks of the form
,...,u u[uJ]
R =
R
so that
and let
a
be
Then there exists
a
is composed of
,
43
(1.6 )
au
=
u
aui+1
=
ui+1 + u
au.
CuJ
where
f
p .
f
j+l = p , then
If
the maximal ideal
If
m C R, then
We may assume that
form (1.4) above.
u Aas follows:
'1
a
f
= 0
RG ->R
j+l < p
is
and the action
ramifies only at
and
f'3/
.
is composed of blocks of the
Within each block, choose new coordinates
Let
=U
u
=ui+j-1 + h
=U i+j-2 + hi+j-1 + (a-l)hi+j
i+j-1 = (a-1)ui+
(1.7)
,...,un)
+ ui.i-l
i+j
u
within the block is linear.
Proof:
(u
has order > 2 and the block size = j+l
E R
at most
=
i+j
+
i+J-2 = (a-l) u i~
Ui
=
(a-l)Jui
=
u +hi+l + (a-l)h2 +...
+ (a-l)~ hi+.
The Jacobian of the map
u
identity matrix, so the
for R.
f
(o,...,0)
is the
Is define a coordinate system
Setting
= (a-l)5i
we have
at
== h + (a-l)h
+ (a-1)2hi2 +
+ (a-l)h
,+4
a i+1
i+t +
ui+1-1
0 <
for
<j
and
au.
Thus
a
has the required form.
By lemma (1.3),
> p
+ if
U.
no block of
a
can have dimension
If the dimension of a block equals
.
ui+j .
(a-l)Jui+j = (a-l)
f
If
,
then
u
=
Consequently,
= (a-l)ai = (a-l)Pui+j = (aP-l)ui+j = 0 '
and so the action is linear.
element
p
Note that this forces the
to be fixed.
a
RG-R
be invariant.
ramifies only at
This implies that
m
f
,
no element
aA
can
/ 0 and hence that
the block size is strictly less than p.
//
The proposition shows that all the non-linear
contributions in
a
can be pushed to the top of each
Jordan block, leaving the remainder of the action in linear
form.
Now suppose that the action of
i.e. at least one
f. / 0
.
a
is not linear,
In this case, the process of
forming successively higher powers of
(a-l)Lui+j
used in
defining the new coordinate system (1.7) can be continued
within each block until the result is zero.
with notation as in (1.6), set
In particular,
45
= f. = (a-l)j+ 1 u
f
(1.8)
f2 = (a-l) j+2 u
= (a-i±)j+ui
Because
= 0
(a-l)
is possible that
,
f1
the element
a
fP J
is zero, but it
i
becomes zero earlier. Let m be
the last index for which
of
.
0 .
fi'
Then the full action
within the block (1.6) can be represented by
af. = fm
i
i
f
(1.9)
1
au
aui
where
f
E R
1
= f
= u
i
2
+ f1
+ f
=u
+ ui+j-1
has order at least 2.
the extended form for the action of
We will call this
a
.
If the action is
similarly extended within each Jordan block, we obtain a
set of power series
(fj%}yj
in
ul,...un
Now consider the power series ring
k[[u
...
un,...w
,..]]
, where one variable
added for each power series
form for
a
.
The ring
defined by letting
variables
w4
a
S =
S
f
w4
is
occurring in the extended
has a natural linear action
act as in (1.9) with the new
replacing the
f4
.
Moreover, the ring
46
with its non-linear action can be
R = k[[U.,...,un]]
obtained from
S
t: S - R
by the map
defined by
$(u ) = u
'
t(w(l) =f((ul,...,un
which is compatible with the group actions.
thus exhibits
R
as a non-linear slice of
The map
4
S .
Rather than compute the invariants of the inhomogeneous action of
a
on R , we instead study the
Because
invariants for the ring S.
on S, the invariant ring
stand than
RG
is often easier to under-
SG
despite its increased dimension.
remains to study how information from
information on
acts homogeneously
a
SG
It then
passes to
RG
The remainder of this chapter is concerned with the
simplest case of studying invariant rings by slicing -namely the case that
that the element
f
single power series
§2.
has only one Jordan block and
of (1.9) is fixed, so that only a
f
appears in the extended action.
The method of non-linear slicing.
Let
of
a
R
R = k[[u 1 ,...,1un
of order
p
and let
defined by
a be an automorphism
47
au
(2.1)
=
+
U
au2 = U
f
+ u
2
aun = un + un-1
where
G
f E R
has order > 2 and is invariant under
denote the
p-cyclic group generated by
a
a . Let
By
.
Proposition (1.5) the dimension n of the ring R is at
most
p.
If
is linear.
n= p
the element
,
f
is zero and the action
In this section we develop a procedure for
studying the non-linear actions arising if
quently, we assume
n < p-l
The ring of invariants
f/O .
Conse-
and we will also assume
RG
is a complete
n > 2.
n-dimensional
local ring whose structure varies with the choice of
f
in (2.1).
Lemma (2.2):
The map
if and only if
some
f
is unramified in dimension > 0
RG-R
contains a term of the form
ui
for
i
Proof:
By Proposition (1.1) the map is unramified away from
ma R
the maximal ideal
r = (au -u ,...,aun-un)
r =
defined above,
exactly when
f
if and only if the ideal
is
m-primary.
(f'u*..,n-
1
)
For the action
and so is
contains a term of the form
Therefore the structure of
R
a
m-primary
un
breaks up into two cases:
48
Proposition (2.3):
the automorphism of
(i) If
f
RG -+R
R = k[[u ,...,un]]
Let
R
and let
a be
defined by (2.1).
contains a term of the form
u
ramifies only at
RG
m
and so
,
then
has an
isolated singularity at the origin.
If
(ii)
contains no
f
uI
term, then
q = (u1 ,..., un 1
along the dimension 1 prime
and
RG
is singular along the image of q,
provided
Proof:
ramifies
RG -+R
n >2 .
Statement (i) follows directly from Lemma (2.1)
u'
n term, then
is equal to the prime ideal
If
and Proposition (1.1.2).
r = (f,u1,...,un- 1 )
(ul,...,un-1).
f
has no
The result again follows from Proposition
(1.1.2).
Now consider the (n+l) -dimensional ring S=k[[uOu,...,unII
The
p-cyclic group
generator
a
G also acts on
S
by defining the <tnen eA
to be
ruO
=
u 0
au1 = u 1 + u
0
aun = un + un-1
The resulting invariant ring
SG
dimensional ring and the map
SG ->S , like the map
has degree
p
is a normal
(n+l)R -+R,
and is ramified only in low dimensions.
49
Proposition (2.5):
SG -+S
The map
p = (u0 ,u ,e.un-
prime ideal
is ramified along the
1) .
SG
Hence
along the 1-dimensional locus determined by
Proof:
.
S
=
The result now follows from Proposition
(1.1.2) and the fact that
The ring
p, = p n sG
r = (au0 -u0 ,...,aun-un)
The ramification ideal
(u0,ul...,*un-1)
is singular
dim S > 3
maps to
R
*(u0)
=
//
.
by the homomorphism
t: S--+R
defined by
f(u 1 ,...,un)
(2.6)
= u.
(u.
l
which assigns to
$
Because
R
,
u0
for
i > 1
1-
the invariant power series
f E RG
is compatible with the group actions on
there is an induced map
S
and
p: SG -+RG between the
invariant subrings.
Now
*, being a map between regular power series rings,
is easily understood:
generated by
(u0 -f)
it is surjective with kernel
.
However, the map
subrings is far more complicated.
Lemma (2.7):
Proof:
u0 E SG
SG
and so cannot
p .
The element
Because
on the invariant
For example, the element
(u0 -f) does not, in general, lie in
generate the kernel of
p
u0 -f E SG
if and only if
it suffices to show that
f= 0.
f
lies
50
in
f
SG
only if it is zero.
By definition, the power series
contains no terms involving
be the first index such that
u . Then
af
u0
f
If
.
f / 0, let
i > 0
contains terms involving
contains terms in
u il
and so does not
equal f .
To derive results for
cp
*
introduce an addition hypothesis
for
on
it is necessary to
analogous to the statements
.
S
Definition (2.8):
An integral domain
R
is factorial if
every height 1 prime of R is principal.
Theorem (2.9):
let
R*
(Samuel, [13])
denote the group of units in R.
cyclic group of automorphisms of
RG ->R
ring
Proof:
R
RG
is factorial if and only if
Let
Let
G
Then the invariant
H1(GR*) = 0.
a be a generator of the cyclic group
H (G,R*)
be a
such that the map
is unramified in codimension 1.
recall that
(2.10)
Given a factorial ring R,
G
and
is defined by
H (G,R*) =
{elements of R* with norm 1)/{image of (a/id)R*)
Note that this is the multiplicative form of (1.2.12).
the set of elements of
R*
Now
with norm 1 can be identified
51
with
R* n Im(a/id)R
where
,
Im(a/id)R c Fract R
argument as in the proof of Hilbert's Theorem 90.
will use this alternative formulation of
Assume
prime of
H (GR*) = 0
RG
and let
The primes of
.
R
p
by an
,
We
H1 (GR*) .
be any height 1
lying over
height 1 and are therefore principal, since
p
R
are all
is factorial.
Consequently, their intersection is given by an ideal (a)
for some
a E R
aa = ta
Therefore
vanishing of
hence
in
ca/a
=
a/u E RG.
R
for some unit
H (GR*)
au/u = t
that
and this ideal is
,
and so
,
t E R*
follows that
u E R*
factorial and let
a unit in
map
RG -4R
a = ub
u E R*
b
(b)
a E R .
,
R*
RG
is
n Im(a/id)R
We need to show that
t = au/u
such that
(a) equals
(a)
(aa)
to
Since
.
t
is
and so is
RG
gives a
by the factoriality of
generates the ideal
(a) in
RG
R
because the
is unramified in codimension 1. Therefore
for some unit
and hence
be any element of
The restriction of
principal ideal
Furthermore
t
R , the ideal
G-invariant.
p
is principal.
for some
there exists a
and
This element must generate the ideal
p
t = aa/a
.
such
a/u = ca/au
To prove the converse, assume that the ring
Then
G
By the
.
there exists a unit
It
.
invariant under
u E R*
.
Also
aa = a(ub) = a(u)-b
t = ca/a = au/u.
Using this criterion for factoriality, Fossum and
52
Griffith have shown
Theorem (2.11):
(Fossum and Griffith, [8])
is factorial if
n = p-i.
The ring
SG
We postpone the proof of this theorem until §3, where the
S = k[[u u ,...,up-l]]
case
is treated in greater detail.
2 < n < p-1
The analogous result for
is not known and
we state it as
Conjecture (2.12):
The ring
SG
is factorial for
2 < n < p-i.
Under the assumption that
SG
possible to obtain results for
Proposition (2.13):
Proof:
ker cp
=
s E S
((u 0 -f)s)S G
a(u0 -f)
=
is factorial, then
such that
(uo-f)s E SG and
af-f
lies in
(u0 -f) under
$
a
u0 - af
u-
in
SG
Consider the image of
The element
analogous to those for
If the ring
there exists a unit
hence
cp
is factorial, it is
f
-
ker $
(af-f)
because
R , and therefore it tas the form
f
(uo-f)t
is invariant
for some
53
t E S
Consequently,
.
a(u0 -f) = (u0 -f)
(u0 -f)t
-
= (u0-f)(1-t)
Now
N(u0 -f) = N(a(u0 -f))
has norm 1.
= N(u 0 -f)N(l-t)
Since the ring
SG
S*
s E S
this choise of
s ,
T((u
Clearly
such that
as/s
=
(u 0 -f)s
.
lies in
SG .
follows from the fact that
and
x
s
is a unit:
and
s
are fixed by
y E S
1y
E SG
such that
because both
w E Im cp
of the same order as
Proposition (2.14):
Given any
x= (u9-f)y.
a
the inverse image of any element
wt
For
given any
It follows from this characterization of
elements
.
The opposite inclusion
x E kercp, there is an element
(uo-f)s
(1-t)
=
(u 0 -f) (1-t) (1-t)~ s
((uo-f)s)SGckercp.
y)
Therefore there
.
=
(u -f)s
x = (u -f)s(s~
S
vanishes,
= (u0 -f)(1-t)as
0 -f)s)
Thus the element
Then
H1(G,S*)
is the group of units of
exists a unit
1-t
is factorial, Theorem (2.9)
implies that the cohomology group
where
and hence
ker cp
that
includes
w
w E RG
which lies in the
54
cp(wt)
=
w
ord w' = ord w
and
.
be given as in the preceding proposition.
s
Let
is 1, so that
s
We may assume that the constant term of
s
such that
We first establish the proposition in the case
Proof:
w = f
w' E S
p , there exists an element
image of
has the form
+ s0 + h
s =
where
s ,h E S
and
consists of all terms of
s
ord(f) - 2
having degrees 1 through
.
s
Then
a((u 0 -f)s) = (u 0-f)(l+ as0 + ah)
= u0 +uo*-as
+ (terms of degree > ord f)
0
On the other hand, the element
a
(uo-f)s
is invariant under
and hence
a((u 0 -f)s) = (u0-f)(1+s
0
+h)
= U0+u s0 +(terms
It follows that
t = s(l+s
under
0
)~
so = as 0 . Now replace
s
Since
so
= 1 + h(l+ s0 )
a , so is
of degree > ord f)
(u0 -f)t
.
Moreover,
by the unit
is invariant
(u0 -f)t
has the
form
u
We claim that
of
f :
+ (terms of degree > ord f)
u - (u0-f)t
The element
fl
.
is the desired preimage
is invariant in
S
fl
because both
55
u0
and
CP(u)
are invariant.
(uo-f)t
f
=
(u0 -f)t
and
lies in
construction,
ord fV > ord f
equal because
ord x < ord p(x)
Also
p(f') = f , since
ker p
.
and so the orders must be
w' E SG
x E S
for all
w E Im cp , and
Now consider the case of a general
let
Finally, by
be any element in its inverse image.
only check that the condition on
ord w' < ord w .
Certainly
all terms of
ord w'
If
We need
can be met.
ord w' < ord w
then
w'
of lowest degree are divisible by u, ;
any term involving only u1 ,..., un would map to itself
e and so the order would not increase.
under
w'
Therefore
has the form
w' = u0 w0 + w
where all terms of
ord T > ord wt
for
w'
.
have degree = ord w' - 1
w0
Replace
u0
and call the result
maps to
w
since
element
w"
by
w"
f'
cp(uo) = p(f')
is invariant.
geneity of the action of
in the expression
The element
.
f
and
.
w"
also
Furthermore, the
This follows from the homoa
on
S
w0
The elements
w'
are invariant, hence so are
and
w
f'
is invariant by construction, the element
.
u0
and
Thus, since
w" = f'w0 +
G0
lies in
Now
S
ord w"
is strictly greater than
ord f' > ord u0 = 1.
ord w'
Therefore this procedure gives a new
invariant of strictly greater order mapping to
Continue as above until
because
ord w
is reached.
w
4 is
Finally, we examine whether the surjectivity of
inherited by
cp
[ 1]
Lemma (2.15):
Given a map
cp: R -+S between
noetherian rings,
(i)
cP: Rp - Sp
If
p
such that
is
an isomorphism for all
depth (R
)
0 , then
=
c
primes
is
injective.
(ii)
If
S
all primes
cp
Proof:
p
such that
is an isomorphism for
depth (R ) < 1 , then
is bijective.
(i) Assume that there exists an element
cp(x) = 0 .
that
p
is a domain and
for all primes
primes
element
Then the image of
p
p E Ass(R)
x
such that
.
depth(R
pp
Since
is already 0 in
R
in
x
)
=
x E R
Sp
0
,
is zero
i.e. for all
is an isomorphism, the
Therefore, for any
.
p E Ass(R)
there is an element
x-r, = 0
This implies that the annihilator of
.
such
r
not contained in any of the primes
E R-p
for which
p E Ass(R)
.
x
is
Therefore
x= 0.
(ii)
map
It follows from part (i) of the lemma that the
R - S
is an injection and so
Furthermore, setting
(0)
,
is also a domain.
the hypothesis
RP'
and
S
are birationally equivalent.
be any element of
S
and consider the ideal
implies that
s
p =
R
R
Sp
Let
57
I = {r E R| cp(r)-s E Im cp
R
This ideal is non-zero because
field of fractions.
r
Let
and
.
S
have the same
be any non-zero element of
I
and consider the diagram
-- +
R/(r 0 ) -Since
R
and
S
S
S/(cpro)
are domains, the elements
are not zero-divisors.
Therefore the map
r0
and
pr0
R/(r0 ) -> S/(cpr 0 )
satisfies the hypotheses of (i), from which it follows
that it is injective.
an
x E R
such that
image 0 in
S/(cpr )
0 )cp(t)
= cp(x) =
i.e. the map
cp
Theorem (2.16):
Then the map
f = cp(u 0 )
Proof:
ep(x)
=
cp(r
0)-s
.
t E R
p(r0 )-s
for which
.
the image of
S
has
x = rot.
But then
s = cp(t) ;
is surjective.
Assume the invariant ring
cp: SG -+ RG
S
x
R/(r0 )
This implies that
SG
is factorial.
is surjective if and only if
contains a term of the form
Let
The element
and so must already be 0 in
Hence there exists a
cp(r
r0 , there exists
By the choice of
un
denote .S /ker cp , which is isomorphic to
under
cp .
along the dimension 1 prime
singular along its image
Recall that
SG -+ S
p = (uo,...,un-1 )
p' = p n sG
and
ramifies
SG
is
(Proposition (2.5)).
58
By Proposition (1.2.14), depth SG = 2 + dim(S/p) = 3
because
a
S .
acts linearly on
-
principal, the depth of
Since
ker p is
is 2.
Consider the commutative diagram
S
S
SG - S
Because
-
G-
R
-- 7S
ramifies along
ramifies along the image of
ideal
.
ideal
,
the induced map
under
$
,
3-R
namely the
r = (f'u..,un-1)
We assume first that
u
p
p
f
contains a term of the form
In this case the ideal
m
r
R , and so the map
of
dimension 0.
is primary to the maximal
S -+R
ramifies only in
Therefore in the diagram
R
)
RG
Y
and
S
ramified only at
m
the maps
prime
a
q c
,
are both finite maps of degree p
.
If we localize at any non-mazimal
we obtain a corresponding diagram where the
a
localized maps
and
q
are now everywhere unramified.
It follows that the third map
unramified [ 9
J.
y :
RGG is also
q
R q
Since the degree of the map is one, it
is an isomorphism.
Therefore the map
7
q
--RG
a
is an isomorphism for all
59
q c3
non-maximal primes
.
depth 7 = 2
As noted above,
and so the set of non-maximal primes includes all primes
of
7
for which depth
implies that
-q
S ' RG
7
and hence that
p
f
un
f E (u1 ,...,un-1)
contained in
p' .
involves no
and so
term.
ker c = (u0 -f)S
p, and
is again a non-maximal prime, call it
Spec S -Spec
SG
In this
n SG
is
p'
to
is
7
If we localize the map
at the prime
p' , the result is again
p-cyclic quotient which is now ramified only at the
closed point.
ring
9
p
cannot be normal:
violates the
that
Since
9
RG
is
By Serre's criterion, the
if
ht
> 1
,
the ring
S 2 -condition, and if
ht p
=
1
is singular along
p
violates the
is a normal ring, the map
The case
SG
p
By Corollary (1.2.13), the depth of
depth S- = 1.
2 and hence
%3.
is surjective.
Therefore the restriction of
singular along this prime.
a
By Lema (2.15) this
.
-
Now assume that
case
q < 1
q
p
,
the fact
R1 -condition.
is not surjective.
n = p-1
In order to develop a normal form for the automorphism
a
in the case that
R
has the maximal dimension
p-l
we first examine which coordinate changes for a general
will preserve
a's
partially-linearized form.
,
R
6o
Proposition
3.1:
such that
R
Let
1 < n < p-l.
of order
p
R = k[[u ,...,un]
Let
a
for any
n
be an automorphism of
defined by
au 1 = u1 + f(u ,.*.un)
ru2 = u2 + u
(3.2)
aun = un + un-1 I
where
a .
f E R
has order at least 2 and is invariant under
Then a change of coordinates
the form of
a
U
(3.3)
a
=c 1 u
2 =
ci E k,
(a-i)nh E RG .
series
'
=
a5
(3.4)
+ c 2f +(a-1)n-lh
2 u1 + c 1u 2
R
cl
#
+ c3 f + (a-i)n-2 h
0 , h E R
+ Ciun
+ h
has order at least 2, and
For this coordinate change, the power
-
5.
=
Proof:
preserves
given above if and only if
un =cnul + cn-lu 2 +...
where
{u>-'-- { 5 j}
is given by
cif + (a-l)nh
The most general form of a coordinate change in
is given by
u
= a1 u
+ a 1 2 u2 +...
un = anlul + an 2 u 2 + ..
+ alnun + g (u ,...,un
+ annun + gn(u,...,un)
61
where
a.. E k
are such that
has order > 2.
|
detla.
/ 0
and
g. E R
If the coordinate change preserves the
form (3.2), then
a
=y
1
1
-
a 1f + a12u1 + a
u2 +... + alnun-1 + ag
must have order at least 2 and so
a 1 2 = a1 3 = 0..
Since
a,
Then
detlai
,
the element
c1 u1 + gi.
U1
c f + ag
/ 0
-
g,
- g1
= a ln= 0.
0 ; set
a1 1 =c .
Furthermore, the expression
must be invariant because
a2 -
The requirement that
2 =
1
T E RG
imposes the
condition
u 1 = c1 u 1 + g 1 = au - u
2
2
= a2 1 f + a2 2u1 + ... + a2nun-l + ag ~
2
Setting the linear terms equal, it follows that
and
a 23 =
24 =
*
= a 2n = 0
As for the
.
c1 =a
htigher
degree terms, this requires that
g, = a2 1 f + ag ~ g
2
2
Set
a2 1 = c 2 *
In general, setting
aU. - U. = U _
c
a
the requirement
,
imposes the conditions
a.
" ij = ci-+
i_-j+l
if
j< i
a..j = 0
if
j > i
lai
and
=
gi_1 = c f
+
(a-l)g
.
22
2
62
gn = h . Then
Set
gn-1 = Cnf + (a-l)h
gn-
= Cn- f + (a-1) (cnf + (a-l)h)
2
= Cn-lf + (a-1)2h ,
(a-l)f = 0 , and in general,
since
(a-l)n-ih
gi = ci+1f +
Finally, note that
I = c 1 f + ag -g 1 = c 1 f + (a-1)nh
Therefore, if
in
RG
f
is fixed, it follows that
cl / 0 , then
If
detfa
| =
and so this defines a change of coordinates for R.
Now consider the special case that
this case
(a-1)n
(a-1)
condition
n = p-l.
In
becomes
-
ap
1
+
ap-
which is the trace operator on
h E R
lies
Combining the above restrictions, we obtain the
.
set of equations (3.3) above.
cl / 0
(a-1)nh
(a-1)nh E RG
2
+
...
+
a + l
,
R . Therefore the
is trivially satisfied for every
and the expression for
? becomes
f
=
c1 f + tr(h)
Thus the power series
f
can be modified, under a suitable
(3.5)
63
coordinate change, by the trace of any
order at least 2.
element of
R
RG
Consequently, power series in
having
which
differ by traces of elements of order > 2 give rise to
equivalent actions.
Now any power series
an automorphism
a
f E RG
of order > 2 determines
via the action (3.2).
Therefore the
classification of automorphisms reduces to the study of
RG
the invariant ring
modulo the ideal
(tr(h)| h E RG, ord h > 2) .
R = k[[u
1 ,...,u
]3]
To examine the situation for
we pass to the model
S = k[[uO 3u 1 ,...,up_l]]
on which
a
operates by the
linear action (2.4).
The invariant ring
SG
if the coordinate system
coordinates
(3.6)
(v )
V
The fact that the
S
can be more easily understood
(u }
is replaced by the new
defined by
= a up
v.
i=
0,...,p-1l
determine a coordinate system for
follows from the relation
p-i-1
u. = (a-l) p-i-u
=
(-l)P-lt(Pjl1)tu
p-1 =
-
t
t=o
p-1
p-i-1
(-1)P-
- - ( - -
t=o
In this system, the automorphism
(i3
),)C~ip-l =
i+l
a
acts by
i+l
v
.
64
and therefore corresponds to a cyclic permutation of the
variables
v
This action by cyclic permutation of the
.
coordinates of
k[[v ,...,vp_l]]
is one of the few wild
p-cyclic actions that has already been studied (see, for
example,
[8 ]).
M
Let
degree
i
S .
in
M
modules
denote the module of homogeneous forms of
Since
a
are stable under
as
G-modules.
of
Mpm
Let
Fm
Lemma (3.8):
a
(i) If
and so can be regarded
denote the invariant submodule
(v0 vl''
generated by
S , the
acts linearly on
p
Y
i
p-l*)m
,
then
M
is a free
G-module.
(ii)
If
i ,let
p
is a free
Proof:
The module
degree
i
M
i = pm.
Then
Mpm/Fm
G-module.
is generated by the monomials of
v0vl'''
p-1
and the automorphism
sends monomials to monomials.
We therefore wish to
in
examine how the orbits of monomials partition
monomial has fewer than
orbit, then since
fixed by all of
a
G .
p
M.
a
If a
.
distinct elements in its
has prime order, the element is
Because
a
permutes the
follows that the monomial is symmetric in the
so has the form
c(v0 v l'''
vp-1t
v.
v.,
,
it
and
65
j ,where
for some
If
p
J
c Ek.
i , this situation can never arise and so the
orbit of each monomial splits off as a free
If
i = pm , then the elements of
G-module.
are the only fixed
Fm
monomials; the orbits of the others again determine free
G-submodules of
M
./
Using this result, we prove the factoriality of the
invariant ring
SG
stated in the preceding section.
Proof of Theorem (2.11):
By Theorem (2.9) it suffices
to prove that the cohomology group
where
S*
is the group of units in
we may replace
S*
For, given any unit
c E k
u , express
(higher degree terms), so
u
Nu = 1
c = 1 .
Also
in the form
Then
c+h
Nu = c
only if
c
is a
+
pth
au/u = a(u/c)/(u/c)
Im(a/id)S* = Im(a/id)U .
so that
Let
F
denote the subgroup of
units of k[[(vO lv''vp-l)]]
group
First note that
.
with constant term l}
is the constant term.
root of unity; i.e.
S
vanishes
by the subgroup
U = (u E S*
where
H1(G,S*)
U
U
composed of the
with constant term 1.
is filtered by subgroups
U = (l+x E U
such that
ord x > i}
The
,
66
Let
U.
denote the image of
The quotients
Lemma (3.8),
U
U./Ui
U/F
are all free
G-modules by
since
U
p/Ui+i
SMi/Fm
Therefore
in
Ha (G,Ui/Ui)
0
=
if
.
i =pm
for all
j >0
.
By induction,
using the short exact sequences
0
U /Ui+ 1
->
it follows that
Now
U/Ui+1
-+
HJ(G,Ul/Ui) = 0
lim Ul/n 2 U/F
HO (G,U/F)
-+
Ui/U
0
>
i >1
for all
and
j >0
and therefore the groups
also vanish for
j > 0.
Thus
HO (G,U)
'
HO (G,F)
j >1. By the periodicity of the cohomology for
for all
cyclic groups, this also implies that
But the action of
Therefore
G
H1 (GU) = 0
on
F
H (GU)E H (G,F)
is trivial and so
and hence
SG
H1(G,F)
is factorial.
=
0
/7
Lemma (3.8) also enables us to understand the structure
of
S
as a
k-vectorspace.
Proposition (3.9):
The invariant ring
by traces of monomials in
0 1''
Proof:
S
SG
is generated
and by powers of
Nv 0
p-l *
By lemma (3.8), the orbit of any monomial
m
which
67
has
Nv0
is not a power of
any invariant containing
a m = tr(m)
except
distinct elements, so
p
as a term must contain
m
Because this holds for all monomials
.
(v0 v1 *vpl)a
which is fixed for any
,
j , any
invariant can be expressed as a sum of traces and powers
Nv 0 . .
of
Now return to the
NuP
in the
1
coordinate system for
p-1 = Nv0
v0 v1 '''
The element
u.
is equal to
u -system, by (3.6).
S
II (au,)
i=0
Therefore the above
=
proposition implies
Corollary (3.10):
w E SG can be expressed in the
and
1
Proof:
h
is
S
an element of
such that
ord h > ord w
h E S
and
such that
w = tr
We can eliminate from
ord w
is a power series in
By the proposition, there exists an
q E k[[Nup_ ]]
h + q(Nup 1 )
h
all terms of degree less than
since the trace of these terms vanishes, by the
homogeneity of the action of
Thus, in the linear model
SG
q
tr(h) + q(Nup_ ), where
form
NuP
Any
a .
//
S , the ring of invariants
differs from the image of trace only in the existence
68
of power series in
NuP
.
1
We want to show that this
situation is preserved in the passage to the ring
wher'e the action of
the notation
N
a
and
is non-linear.
tr
trace for the action of
R
We remark that
will now denote the norm and
a
on
R , not on
Lemma (3.11): Given any w E RG
G
G
of cp: S ->R , there exists an
S .
which lies in the image
h ER
and
q E k[[Nu
1
]]
such that
w = tr h + g(Nup 1 )
and
ord h > ord w .
Proof:
This follows from Proposition (2.14) and Corollary
(3.10).
Theorem (3.12):
an automorphism of
consist
for some
R of order
p
R
so that
au 1
=u
au2
=u
a
+ (Nup_
2
and let
a be
whose linear terms
of a single Jordan block.
of coordinates for
(3.13)
R = k[[u 1 ,...,u_ 1 ]]
Let
Then there is a choice
has the form
)
+ u1
j > 1. Furthermore, the automorphism
a
completely determined by the choice of the integer
is
j .
If
69
the action of
a
j
Spec R, then
Proof:
is not free outside the closed point of
=
au1 = u
i.e.
co,
We may assume,
and the action is linear.
by Proposition (1.5),
that the action
f= au - u1 .
has been brought into partially linear form with
The proof is divided into two main steps:
we show
first that there exists a coordinate change so that
is
and then that coordinates can be
a power series in
NuP
chosen so that
is a power of
f
f
1
Nu P-1
itself.
The first step proceeds by induction.
It suffices
to show that tnere exists a converging series of coordinate
changes in
NuP
R
under which the terms of
f
not involving
are forced into arbitrarily high degrees.
1
assume that
f
h ER
where
is
expressed in the form
has order
> n
and
q
Therefore,
trh + q(Nuq
1
) ,
is a power series in
We show that it is possible to choose coordinates
- =
mn-1
ui
u (mod m- ) so that the resulting f has the form
NuP
.
1
?= tr
i + q(Nip_)
power series in
ord b > n+l
,where
Nu-
such that
Choose new coordinates
letting
u_
- h.
c
1, c
=
= 0
for
u
f
+ tr(-h)
= tr
q(Napli)
u
=
u
+h, so in the
is a
q(Np
i > 1, and setting
f
1
) (mod mn
p-
=
is given by
h + q(Nu
= q(Nup_1 )
Now
q
as in Proposition (3.1),
Then the corresponding
=
and
1 ) -
tr
h
.
u -system the element
Nu
70
is equal to
N(_
P 1 +h).
i =q(N(fP_
+h))
+ (terms of degree > ord q(Nup_ ) - 1 +ord f)
=
Denote the difference
ord
?
because both
? and
where
q'
ord h > ord 8
in
cp: SG -> RG
)
R E R
and some
q+q' = q
=
q(Nap_ 1 ) + tr
.
Also
,
for
we obtain
nR
q(Nip1 ) - q(N~p_l) = q(Np_l)
order > ord 6 > n+l
and so has
Nu
Setting
.
and note that
do, and so, by Lemma (3.11),
tr h + q'(Np_
=
ord h > n+1
8
1 + ord h > n +1
q(N~p_l)
for some power series
?
-
by
lies in the image of
6
8
q(Np_l)
-
> ord q(Nup_)
The power series
which
Therefore
.
Thus this coordinate
change satisfies the conditions mentioned above and so
this completes step one.
f = q(Nup_ )
Assume now that
q
and write
f
as
f = (Nup-1 )(c
for some
0
+c(Nup_1 ) +c 2 (Nu_1 )2 +...)
j > 0, where
c = c0 + cl(Nup_1 ) +
Let
a E RG
for some power series
...
E k
c
,
and
c0 / 0.
which is a unit in
Set
G
R
be a solution to the equation
71
Mpj-l=
Such a solution exists because
is prime to
u. = au
p
RG
Now consider the map
.
pj - 1
is complete and
ui
u
where
a.
This is an invertible transformation since
.
is a unit, and so defines a coordinate change in
R
Under
.
this change, we have
f
=
=
mc(Nup9j
=
P(Nup_)'
=N(au,,-)j
Hence
f
has the required form.
In fact, given any integer
j
,
the element
f = (Nup_ )j , and hence also the action of
a
,
is
completely determined by the set of equations (3.13).
This
follows from the fact that the identity
p-1i
Nu_
P-
can be used to solve for
u,...,uP_1 .
Ca u
n
i=O
NuP
-
as a power series in
1
Specifically, by expanding
a u
in terms
of the action (3.13), we obtain a polynomial expression
for
Nup_
in terms of
expression for
Thus
Nu
1
NuP
, and
(Nup_)j
.
The
can then be substituted into this
Nu
polynomial to eliminate
a formula for
u ,...,u_
1
NuP
1
recursively, resulting in
in terms of
u ,...,u_
depends only on the choice of
alone.
j.
72
Finally, if the action of
a
on
Spec R
is not free
outside the closed point, then the map
RG -+R
ramifies
in dimension > 0 and so, by Lemma (2.2), the power series
f
contains no terms of the form
+ ...
,this implies that
p-1
action is linear.
U
u i
Since
f = 0
Nu._
=
and the resulting
Thus, by a suitable change of coordinates, the action
of
a
form.
ring
can always be brought into a particularly simple
Unfortunately, the equations defining the invariant
RG
for these actions are, in general, difficult to
write down.
In the case
p = 3, the ring is Cohen-Macaulay
and an explicit equation for the quotient-singularity can
be calculated.
This is given in the next section (see
Corollary (4.15)).
structure of
§4.
RG
The case
However, even when
p
=
5, the ring
is not known.
n = 2.
We now apply the methods of §2 to the case of surface
singularities.
k[[u ,u2 ]]
R
be the power series ring
in two variables and let
automorphism of
(4.1)
Therefore let
R
defined by
au 1 =
u 1 + f
au2
u2 + u
=
,
a
be the
p-cyclic
73
where
has order at least two.
f E RG
for this action is
k[[uO,ulu
2 ]]
,
The linear model
S
with homogeneous action
given by
(4.2)
Here
au0
=
u0
au
=
u
au2
=
u2 + u 1
dim RG = 2 = depth RG
dim SG = 3 = depth SG
+ uO
(by Proposition (1.2.6))
and
(by Proposition (1.2.14)) and so
both invariant rings are Cohen-Macaulay.
In this case it
is possible to write down equations defining these rings
of invariants explicitly.
We begin by examin.,ing the linear model
invariant ring
Denoting
Nu1
SG
The
contains the elements
x
by
x = n a u
S .
Nu2
and
by
=
p-1
R (u1 +muO)
m=0
=
up
1
-
u
0~
u
uO, Nu1 , and Nu 2 '
y , one can compute
1
(4-3)
p-1
y =lau 2 = m=0
U (u2 +mu +(2 )uO)
= u2 - u
Let
S'
be the subring of
Lemma (4.4):
of ranks
p2
The rings
and
S
1
u2 + (terms divisible by u
SG
and
generated by
SG
p , respectively.
are free
)
uO, x, and y.
S'-modules
74
Proof:
uO, x, and
The elements
parameters in
The ring
S
S ; hence
S
SG
S'
S
.
form a system of
is a finite extension of
S'
is regular, therefore Cohen-Macaulay, and so
0 < i,j < p-1 .
,
(u iu)
is finite of rank
p2
A basis is given by the
the extension is free [14].
elements
y
Now the extension
SG
p , so
has rank
Again the extension is free because
p
over
SG is
Cohen-Macaulay.
The element
(4.5)
is
z
=
u
also invariant under
the ring
a
and
z
z
The invariant
(p+])/2
R = zp +
(-1)n
n=2
z
not contained in
uO,u 1 ,u2
S
is two.
over
S'
satisfies the relation
((2n-2)!/n!(n-1) !)u2p-2nzn
+ 2u0y
0
over the ring
is
generates the extension
Lemma (4.6):
Proof*:
2u u2
S' because its degree in
We claim that
(4.7)
-
-uu
-
X2
S' .
The relation
9 is homogeneous of degree
2p
*Thanks to Ira Gessel for discovering this considerably
simpler (and shorter!)
proof.
in
and
u2
and so, for simplicitly, we will verify the
relation in the inhomogeneous form obtained by setting
u
= 1 .
Consequently,
the elements
x ,
, and
z
are
now replaced by their inhomogeneous forms
p-1
(u +m)
y=
z z =
xP
=
U1
m=0
p-1
n
+ mu + 2
m=O (u2
u2
-
1
The coefficients
2u2
'
(2n-2)!/n!(n-l)!
appearing in
the equation (4.7) have a particularly nice representation
in terms of binomial coefficients.
(1/2)
n
(1/2) (-1/2) ...
Specifically,
(-(2n-3)/2)
n!
c
n! (n-1)
(l)en-1
2 2n- 1
Therefore, with coefficients considered
(p+l)/2
(p+D/2
=
n=2
n! (n-1) !
n > 1
(2)(4). . . (2n-2)
(2)(4)... (2n-2)
(-1)n-1 (1)(3) ... (2n-3)
2
for
mod p , we obtain
22n-1 1/2
-
n
n=2
(p+13/2
n 2
n
n=2
=- ( (1 +4 z p+D/2
-
1 -
(p+D/2)(4 z))
=-1((1+4z)(p+D/2 - 1 -2z)
76
Next we develop a suitable formulation of the product
y .
appearing in the definition of
p-1
(+
(p-])/2 _ (_)IP-l/2) 2
+ (se - $) 2
)2
(m+
We first claim that
m=O
To prove this, one need only check that the right-hand
a = -(m+s)2
side vanishes when
for
m = 0,1
.p-1
This is straightforward and we omit the details.
Now setting
a
and setting
=
2u2 -(ul
=
2u 2
=
-z
-
u
- 1/2
, we have
2u
(u1 - 1/2)2 + (m +u
5
=
a + (m+5)
=
=
-
u1
-
1/2)2
- 1/4
+ u
1/4
2
2u 2
+
-
1/2)2
m2 + 2mu 1 -
+ (M)2
= 2(u 2 + mu 1
Therefore
p-1
n
2y =2P
m=O
p-1
u2
+mu
+())
(m+s)2)
1)2
(m+
=0
(-z-1/ 4 )(z
=
/ 4)(P-1)/2
+ ((u
z - 1/4)
=
-z
- 1/4 +
-
(_l)plY2)2
- 1/2)
- (u1 - 1/2))2
+2(z +1/4)(P+1)/2 - z -'1/4 + (u
+1
+
- z -(4z
- 1/4 +x
2
- u1 )2
77
- 1 - 2z) +x 2
= zP + I((1 + 4z)
=x2 -zP -7 (-)n((2n-2) ! /n! (n-1)t)zn
and so the proof is complete.
Proposition (4.8):
The invariant ring
SG
where
S'[z]/(/)
9 is the relation of the preceding lemma.
Proof:
Clearly, we have the inclusions
S [z]/(R) C SG
S'
Both
=
SG is defined by
SG
and
S'[z]/(9)
and the relation
S' [z]/(R)
S'[z]/(P9)
= SG
R
, it
is
are degree
irreducible.
p
extensions of
Therefore to show
suffices to show that the ring
is normal.
Consider the partial derivatives
3P/ x
=
-2x
3R/by
=
2u 0e
These partials vanish simultaneously only along the codimension 2 locus
(x =
uO
=
O}
.
Thus
is
non-singular in codimension 1 and, being a hypersurface,
it is therefore normal.
//
We now examine how this structure carries over to the
ring R .
The ring of invariants
RG again contains
Nu
and
Nu2 , where the norms are now computed with respect to the
inhomogeneous action of a on R . We continue to denote
these norms by x and y ; they correspond to the images of
the above
x and y under the map
f 1 u
x = U-
y = u2
and therefore
-
cp: SG -*RG .
Note that now
up + (terms of degree > p)
uP1 u2 + (terms of degree > p)
x and y alone forma systemof parameters in R.
Consequently, we obtain the following analogue of
Lemma (4.4) :
Lemma (4.10):
of ranks
The rings
R and RG are free k[[x,y]]-modules
//
p2 and p , respectively.
We concentrate on the case that RG -4R
at the maximal ideal
singularity.
CP: SG--+RG
ramifies only
m c R, so that RG has an isolated
If Conjecture (2.12) is true, then the map
is surjective, by Lemma (2.2) and Theorem (2.16).
In this case the invariant ring RG is generated by the
In fact, it is possible
images of x, y, z, and uO under cD.
to show that the image of
RG
over
k[[x,y]]
z alone generates the extension
Denote the element
p(z) = u2
fu
-2fu
again by z
Proposition (4.11):
then RG is
If the map
CP: SG -+RG is surjective,
generated over k[[x,y]]
by the single invariant z.
79
Proof:
Let
w
be any element of the invariant ring
The element
w
can be represented as a power series in
RG
x, y, z, and
f; it suffices to show that all terms
involving
can be forced into arbitrarily high degrees.
f
By Proposition (2.14), there exists an element
wt E SG
w'
such that
in the form
p(w')
=
w
r + u0 s ,where
k[[x,y,z,u0 ]]/(9)
and
r
ord w' = ord w
and
Write
.
r,s E SG=
depends on
x, y, and
z
alone.
Now
w
Therefore
=
p(r+u 0 s)
p(r) + f-cp(s)
=
ord w = ord w' = ord cp(r)
ord(u 0 s) > ord w.
and
Thus the terms in
w
ord(f-cp(s))
which involve
lie
in degrees strictly greater than the order of
Set
w 1 = fecp(s)
.
The element
w1
can repeat the above argument using
lies in
w1
RG
>
f
w
and so we
in place of
w .
Continuing in this manner, we conclude that a representation
for
w
can be chosen so that terms in
f
have arbitrarily
high degree.
The elements x, y, z, and f satisfy the relation
(p+1/2
(-1)n((2n-2)!/n!(n-l)!)f 2p- 2nzn
(4.12) 9' = z +
n=2
+ 2f y - x 2 = 0
GG
in RG , corresponding
to the relation
If the map
by x, y, z, and uO in SG.
c(4.7) satisfied
cp is surjiective,
8o
the preceding proposition implies that
k[[x,y,z]]
among
and, consequently, that
x, y, and
z
alone.
f
lies in
F,' defines a relation
In fact, we now show that
even without the assumption of the surjectivity of
hypothesis that
Lemma (4.13):
x
lies in
k[[x,y,z]]
m C R
The map
RG -* R
if and only if
ramifies only at the maximal
f
By Lemma (2.2), the map is unramified away from
if and only if
f
some
x = u1 - f 1 u 1
i
f
Now
.
contains a term of the form
is relatively prime to
is not divisible by
are
is relatively prime to
z .
and
Proof:
so
is sufficient to
RG = k[[x,y,z]]/(A')
prove that
ideal
f
cp , the
u1
and
u1 .
and
x
and
u2
for
zz~= u 2fu -2f2fu ,
1 -f
1
2
z
exactly when it
Since the only variables in
u2 , this requires that
m
f
R
contain a pure
u2 -term.
Theorem (4.14):
If
f E k[[x,y,z]]
and the map
R
- R
is ramified only at the maximal ideal, then the ring of
invariants
RG
is equal to
k([x,y]][z]/(R')
where
Proof:
R'
is the relation given by equation (4.12) above.
As in Proposition (4.8), it suffices to show that
the relation
Let
cn
=
k[[x,y]][z]
defines a normal quotient of
'
(-l)n(( 2n-2 )!/n!(n-l)!)
and consider the partial
derivatives
(p+1D/2
)-'/3x
=
(-2n)cnf 2p-2n-i zn (3f/x) -
2x
n=2
(p+l/2
2p- 2 n-lzn( f/y) +2
(-2n)cff
=
n=2
(p+])/2
[(-2n)c f2p-2n-lzfn()f/z) + nc n f2 p-2nzn-l
)R'/3z =
Ln
n=2
= fp- 2 z(2c 2 fp-2 + terms divisible by z
)
There are three cases to examine:
(i)
If
(39'/3y) = ()R'/ z) = 0
f = 0, then
(II'/3x)
= -2x
.
The elements
and
f
and
are
x
relatively prime, by Lemma (4.13), and so vanish
simultaneously only at the maximal ideal
(ii)
of
k[[x,y]][z]/(')
If
z = 0, then
2fP
at
(iii)
.
(32'/3x)
=
-2x
and
(3,'/3y)
=
and again these simultaneously vanish only
m' .
The only other possibility for which (3R'/3z) = 0
is that
2c2 fp
generated by
prime and
at
Therefore
m'
2
z .
is contained in the ideal
But
f
and
z
are relatively
c2 /1 0, so again this happens only
m' .
k[[x,y]][z]/(P9')
is singular only at the maximal
82
ideal.
Since the ring is non-singular in codimension 1 and
defines a hypersurface, it is normal.
As in the previous section, we ask how much the
choice of
f
can be restricted.
By Proposition (3.1), f
can be altered by any element of the form (a-1) 2h which
a , provided
is invariant under
h
is an element of
R
of order at least two.
Now
x
x E tr(R)
=
=
Nu
since
N((a-l)u 2 )
=
n
(a u 2 -
i=1
aC~1 u 2)
and it is easily checked that the expansion of this product
can be represented by a sum of traces. Therefore x lies
2
in Im(a-l) R D tr R and so any terms of f involving x
can be eliminated by a process analogous to that used in
the proof of Theorem (3.12).
On the other hand, the element
eliminated from
f
dimension > 0.
if the map
y
RG -- R
cannot be entirely
is unramified in
This follows from the fact'-f
must contain
a pure
u -term (Lemma (2.2)) and of the generators
and
of
z
RG
is the only element which does.
y
,
x, y,
Moreover, one can easily calculate that tnere is no element
h E R
of order
y E Im(a-l) 2
> 2
only if
y = (a-1) 2(u2 + ...
The fate of
for which
f = y + ...
(a-l) 2h = y .
In fact
and then
)
z
is still unresolved.
One would like
83
to be able to eliminate all variables except
dim R = p-l .
as was done in the case
the
z 9 Im(a-l) 2S
case, here
p-l
y
unless
In the special case that
p-l
cp(z)
p = 3 , and we
p: SG - R
does lie in
p = 3
f,
However, unlike
have not yet determined whether the kernel of
is sufficiently large so that
from
Im(a-l) 2R
the dimensions 2 and
,
coincide, and so the results of sections 3 and 4 can
be combined to obtain
Corollary (4.15):
and let
a
be any automorphism of
linear terms consist
map
RG
(4.16)
for some
RG
-*
R
R = k[[u 1 ,u2 ]]
Let
R
,
where
char k = 3
of order 3 whose
of a single Jordan block.
If the
ramifies only at the maximal ideal of
R
j > 0
= k[[x,y,z]]/(z3 +y2jz
2
R, then
_ y3j+l_ X2
If the map ramifies in dimension 1, then
.
is regular.
Proof:
By Theorem (3.12), there exists a choice of
coordinates for
for some
R
so that
a
au1
=
u
au 2
=
u 2 + u1
j > 0, where
y = Nu2
has the form
+ yj
.
If the map
G
R -R
ramifies only in dimension 0, then Theorem (4.14) gives
the desired form for
RG
.
If the map is ramified in
84
j = o
dimension 1, then
and
a
is
given by the linear
action
au
u2
u
The elements
and
=
u
=
u2 + u1
y = Nu 2 =2 u 2 - U1
u22
are invariant
under this action and it is easy to check that they are
algebraically independent and generate the invariant ring.
Thus
RG
is a regular complete local ring.
It is interesting to note that if
j = 1, then the
resulting singularity (4.16) is a rational double point
of type
E6
.
singularities,
However, in the range
1 < j <
though Cohen-Macaulay,
are no longer rational.
co,
the
Finally, we remark that the structure of the invariant
ring
RG
for dimensions between 2 and
understood.
S
p-l
is not yet
For these intermediate rings the linear model
is-a truncated form of the
(p-l)-dimensional model and
no longer has the filtration by free G-modules which makes
the
p-l
analysis possible.
Nor does the invariant ring
have the compensating property of being Cohen-Macaulay
and hence possible to compute directly, as in the
2-dimensional case.
CHAPTER IV
Examples of Generalized Actions.
In the preceding chapter we examined the forms for
2Z/p -actions generated by an element
a
with a single
Jordan block and a single higher order entry in its
extended linear form.
We would like to know how information
about this basic form can be generalized to a broader range
of group actions.
The first section of this chapter
examines several methods of extending the one-block
Z/p-
action results and several conjectures are formulated
about the resulting canonical forms.
In the second section,
these generalized forms are illustrated by analyzing the
actions which give rise to tne rational double points in
characteristics 2 and 3.
§1.
Remarks on generalized forms.
Let
R
be a formal power series ring over an alge-
braically closed field
be a cyclic group of
an element
a
a
.
k
of characteristic
k-automorphisms of
R
p
and let
G
generated by
By Proposition (111.1.5), the action of
can be brought into a partially-linear form in which all
higher-order terms are pushed to the top of each Jordan
block.
Following the methods of Chapter III, §1, the
non-linear action can then be extended into the linear-style
86
action (111.1.9) and a linear model
S
constructed in
which one new variable is added for each non-linear term
in this extended form.
However, if the number of new
variables is greater than one, the methods of Chapter III,
§2 must be modified:
the invariant ring
R
no longer
can be realized as a simple hypersurface slice of
SG
and
the coordinate changes required to bring each of the
higher-order terms into a canonical form must be compatibly
chosen.
In this section we indicate several ways in which
the results of Chapter III may be extended; specifically,
we examine the cases of 2Z/p-actions generated by an
element whose linear terms consist of several Jordan blocks
and of 2Z/p -actions for
(i) Multiple-block
Assume that
and that
a
i > 1 .
2Z/p-actions.
is a
;.;u
,...,u
R=k[[u
p-cyclic automorphism of
linear part consists of
n
R
whose
Jordan blocks, each of the
maximal non-linear dimension
p-l.
By Proposition (111.1.5),
there exists a choice of coordinates so that the action of
a
within each Jordan block has the form
au i.1
(1.1
= u
ui,2
au
1
+ f (ul1 ,... ,unol)
= u i,2 + uil
-
+ uip-2
'
87
i = 1,...,n
for
is
,
invariant under
all of the
where
f
a .
E R
has order at least 2 and
Note that each
f. may involve
uj's, not only those within its block.
In
the one-block case, Theorem (111.3.12) shows that it is
possible to choose coordinates so that the single higherorder term
f
is a power of
Nup_
.
However, in a
several-block action, there is a possibility of interplay
between the higher-order terms of different blocks.
We
therefore conjecture that Theorem (111.3.12) has the
following generalization:
Conjecture (1.2):
For
R
and
a
a choice of coordinates so that
where
q
j = 1,...,n
f
as above, there exists
= qi(Nu
is a power series in the norms of
.
Furthermore,
if
the map
RG - R
only at the maximal ideal, then the ideal
primary to
(u
,...,un,p-1
In the case
n = p
established by Artin [2
=
2
J. A
),
,.,Nu
u
for
ramifies
(f ,...,f )
is
*
,
this result has been
non-linear action in
characteristic 2 has Jordan blocks of maximal size 1, so
that actions in dimension 2 have the form
au
= u
+ f
(1.3)
au2 = u 2
where
f
= q (Nu ,Nu2 )
+
.
f2
'
If the action is free except
88
at the closed point, then
maximal ideal
m = (u ,u2)
relatively prime.
(f ,f2 )
is primary to the
and hence
f
z = u 1 f2 + u 2 f1
z2 + ff
(1.4)
This action is
2z
,
x = Nu1
is
f
= x
and
are
,
y = Nu 2
subject to the single relation
+ f2y + f2 x
=
0
one of the basic forms arising in the study
of rational double points in characteristic 2.
if
f2
In this case the resulting invariant
ring is generated by the three elements
and
and
For example,
f2 = y , the resulting quotient-singularity
defined by the equation
(1.5)
2
2
2
z + xyz + xy + y
which is a double point of type
If the Jordan blocks of
0
D4
a have less than maximal
dimension, the situation is far more complicated.
As the
size of a block decreases, the number of choices for
higher-order terms increases, so that the single-block
actions themselves have a greater variety of forms.
Nonetheless, examples based on 2-block actions where the
action within each block is one of the many one-dimensional
forms of Chapter II indicate that the analogue to
Conjecture (1.2) still holds:
the main principle in
generalizing from one- to several- block actions should
be to allow higher-order terms to be chosen from the common
89
pool established by the choices from each of the one-block
components.
(ii)
2Z/p -actions for
Now assume that
series ring
i > 1
a
is an automorphism of the power
R = k[[u 1 ,...un]]
of order
p
for
i > 1
For simplicity, we will assume that the linear part of
consists of a single Jordan block.
Z/p -actions,
the action of
coordinates for
a
As in the case of
can be chosen so that
has the form
au 1
(1.6)
R
a
=
U1
+
f(u,.*,u
n)
au2 = u2 + u
aun = un + u n-l
where
f E R
has order at least 2.
The maximal size for a Jordan block in a linear
action is
p , so that the maximal linear model is given
S = k[[v 1 ,...,v ]]
p
by the power series ring
with
automorphism
(1.7)
2/p -
av
-
av2
=
av i=
p
2 + v
v
+ v
p
.
p -l
Any non-linear one-block action can be realized as the
image of this model under the map
t: S -> R
defined by
90
(a-l)p -n+j (f)
4(v)
(1.8)
=
for
j = 1,...,p -n-i
f
for
j = p -n
u n-p i +j
for
j = p -n+1,...,p
Note that this map simply sends the last
S
to
f
,
u 1 ,. . .,un , respectively,
n+l
letters in
and the images of
the earlier letters are determined by the action of
a
on
f.
a
on
R
(If the extended linear form for the action of
has length less than
several
p
, then the images of the first
v.'s will be 0.)
In order to employ the methods of Chapter III in
analyzing canonical forms for 2Z/p -actions, we consider
Pi-i
the 2Z/p-subaction on S generated by T = a.
The action of
T
=
Jordan blocks of size
p
.
Lemma (1.9):
p
a
on
S
splits into
The action within each
block is defined by
= v
Tv
1-
i-1
j+(p-1)p
where
Proof:
j = 1,...,p
+ v-
4+(p-l)p
-
+
j+(p-2)p
i-i
This follows at once from the definition of
using the facts that
T-1
=
ap
-
1
=(-1
a
,
l
and
v.
,
~Jv
(C-l)
=
p//
Therefore the linear model
generated by
T
with 2Z/p-action
closely resembles the linear model for
the multiple-block
Jordan blocks.
S
2Z/p-action consisting of
p i-
maximal
However, in the multiple-block case the
blocks remain independent in the passage from linear to
non-linear systems, whereas in the 2Z/p -action the blocks
T
are intertwined, the elements of the same block of
being distributed to every
(p i)st
position in
Consequently, the single higher-order assignment
completely determines the images of all earlier
S
*(v i)
p -n
v 's.
=f
We are primarily concerned with the case that the
is unramified in dimension > 0 and so now
-. R
R
map
examine which elements
order terms in
R
v.
in order for this to be true.
Lemma (1.10):
If the map
maximal ideal,
then
Proof:
RG -- R
n = dim R
If the action of
ideal of
must be assigned to higher-
G
G .
at most
u.
R
cannot fix any
of
T
maps to a non-linear action on
i:
S -R
- p i-
can be fixed by any
In particular, the induced action of
on
that
p
is free except at the maximal
R , then no coordinate
subgroup of
is
ramifies only at the
u
T
and so each of the Jordan blocks
R
.
This implies
sends at least the top letter in each block,
92
namely
v1 ,...,v il
the remaining
to a higher-order power series in
,
v 's.
Hence at most
p -p
variables
* .
remain degree 1 under
In the maximal case, i.e.
the top letter in each block of
R .
order power series in
dim R = p -p
T
only
,
is assigned to a higher-
The situation is therefore
like that of the preceding example where now the lowest
letters in each block are given by
j = l,...,p
the map
in
=
,for
i
fli u
u
il
= 1,...,p
j
p
.
+j
p -2p
denote the norm of the element
respect to the subgroup generated by
Under
above.
j
u.
t , these elements map to the last
R , namely
N u.
in place of the
,
v
letters
Let
u
with
T ; note that this
is not, in general, invariant under the full group
G .
Then by analogy with the multiple-block forms we obtain
the following conjecture for the maximal 2Z/p -action:
R = k[[u 1 ,...,u
]
and let
p -p
whose linear
a be an automorphism of R of order p
Conjecture (1.11):
Let
part consists of a single Jordan block.
a choice of coordinates for
is defined by
R
Then there exists
so that the action of
a
93
Ou 1=
u1 + q
au 2
= U2 + u
au
=u
p -p
where
for
q
j
=
+ u
p -p
p -p
-l
is a power series in the elements N u
+j
p -2p
G
i-1
.
Furthermore, if R -4R ramifies
1,...,p
only at the maximal ideal, then
N u i il
p -p
involving
q
contains a term
alone.
As in the case of multiple-block 2Z/p-actions, the
simplest forms of these 2Z/p -actions arise in the classification of the rational double points.
p= i= 2,
then
a
is a
2Z/
4
For example, if
-automorphism of
k[[u ,u 2 ]
defined by
au
(1.12)
=u 1 + f
1u
Cu2 = u2 + u 1
If
f = NTu 2 = u2 (u2 +f)
of
u2
then
f
can be expressed in terms
alone by solving this equation recursively.
resulting invariant ring
RG
is generated by the three
elements
x
=
y =
f-af
af
z = f 2u
-
f
+ f u 2 + f af + f'af.u
subject to the single relation
The
+ (af) 2u 2
94
z2 + x3 + xy3 + xyz
=
0
.
This equation defines a double point of type
E
.
It is
interesting to note that the 2Z/2-subaction generated by
2
T = a
is, after a suitable coordinate change, precisely
the multiple-block 2Z/2-action leading to the rational
double point (1.5) of type
D4
considered in part (i)
above.
§2.
The rational double points.
In this section we illustrate the variety of actions
possible over fields
k
of non-zero characteristic by
examining the automorphismof the smooth scheme
X = Spec k[[u,v]]
which yield the rational double points
as quotient-singularities.
rational double
points are obtained as quotients of
by the finite subgroups
space with basis
of
In characteristic zero, the
u,v .
G
of
acting on the vector
SL2
However, if
G , the group action degenerates
singularities in characteristic
p
simply to the smooth reduction of
action.
X
p
divides the order
mod p
and the
do not always correspond
X
modulo this group
Artin [ 3 ] has shown that the rational double
points in characteristic
p
do still
have coverings by
smooth schemes; we now calculate the actions necessary to
exhibit these singularities as quotients of the form
up to purely inseparable extensions.
X/G
,
95
Unlike in characteristic zero, the rational double
are not necessarily rigid, i.e.
p
points in characteristic
there may be several non-isomorpnic singularities having
the same configuration of exceptional curves in their
The classification of the double
minimal resolutions.
points used below is
that given in [ 3 ]-
The simplest family of singularities are those of
type
n > 1 , which are defined in all charac-
An , for
teristics by the equation
=0
zn+1 - xy
(2.1)
In characteristic zero,
quotient of
An-
the singularity
by the cyclic subgroup of
X
is the
SL2
generated by
C0
(2.2)
where
ring
1
C
k[[u,v]]G
is generated by the three elements
,
=
x = un
(2.3)
subject to the single relation
If
pjn
,
zn
z = uv
=
xy
then the covering of
by (2.3) becomes inseparable
m
The invariant
root of unity.
nth
is a primitive
is relatively prime to
Let
by
X
defined
n = p m
where
p . Then the substitution
e
x =x
mod p .
An-
e
,
yy-
,
defines a purely inseparable cover of
z=z
An- 1
by the simpler
singularity
Am- 1
The singularity
quotient of
X since
.
Am-
pm , and therefore
realized as the quotient of
is
a tame
An-1
can be
X by a tame
m-cyclic action
e
plus a purely inseparable extension of degree p . Tnis
is one of the few cases where the characteristic zero
behavior specializes nicely to the characteristic
singularity.
p
We will see below that in general such a
simple explanation is not possible.
To analyze the remaining singularities we first
G
observe that if the group
producing the characteristic
G/H
zero singularity has a quotient
p , then the cover
to characteristic
X/H
p .
-
X/G
of order prime to
is preserved in the passage
Thus the singularity
explained in terms of the simpler singularity
X/G
X/H
can be
and
so we need only consider those cases in which the characteristic zero group has no quotients of order prime to p
This occurs only in characteristics 2, 3, and 5, which we
examine separately below.
(i) Actions in characteristic 2.
The actions producing the rational double points in
characteristic 2 are modifications of the basic forms (1.3)
and (1.12) described in §1.
Of particular interest are two
generalizations of the multiple-block 2Z/2-action (1.3):
First consider the effect of allowing the higher-order
terms
f1
and
f2
to take on common factors, say
f
=
ac
f2 = bc
and
y
,
Nv , and the elements
=
prime.
c
a, b, and
where
The action of
a
x = Nu
are power series in
a
and
b
are relatively
is then given by
au
=
u + ac
av
=
v + bc
(2.4)
and the invariant ring is generated by the three elements
z = ub + va , which satisfy the relation
x, y, and
z2 + abcz + a2y + b2x
(2.5)
Because of the common factor in
f
0
=
f2 , the covering
and
of this singularity by the smooth scheme
(c = 0) .
along the locus
X
Note that if
c
is ramified
=
1, this gives
the original unramified form (1.4).
Secondly, the action (1.3) may be modified by allowing
Nu
and
Nv
to be polynomials in the invariants
y , instead of
x
y
and
of replacing the
themselves.
and
x
y
polynomial expressions for
the smooth cover
x
and
This has the effect
in equation (2.5) by the
Nu
and
Nv
and of replacing
by a more complicated surface.
X
To illustrate these actions, we first examine the
singularities of type
DN
.
These singularities are
described in characteristic 2 by equations of two forms,
depending on whether
(2.6)
Dr
2n
D2n
:z
2
N
is odd or even:
+ xyn-rz + x2y + xyn
=
0
2 + xyn-rz + x2y + ynz
=
0
(O < r < n-1)
98
In characteristic 0, the singularity
quotient of
DN
is obtained as a
aN-2
X by the binary dihedral group
In
.
characteristic 2 the actions producing these singularities
split into several types depending on the relation between
r
and
n.
If
N = 2n
and
n > 2r
then the singularity
,
X .
has a ramified double cover by the smooth scheme
n
The action for even
DN
is of the standard form (2.4):
n
CU = u + xy
(27)av
where
= v + yn-r
x = Nu
and
y = Nv .
The ramification in the cover
y(n/2)-rin the higher-order
is seen in the common factor
terms.
The equations defining the covering of
can be obtained by expanding the identities
DN
x = Nu
by
X
and
y = Nv; specifically
r
7-n
u
u =x
+ xy
v2 +yn-rv
If
n
=y
is odd, the singularity can be realized as a quotient
by the related action
n+l
au = u + xy
(2.8)
where
av = v + yn-r
y = Nv
but now
Nu = xy
The equation
.
n+1
Nu =xy =u
2
+ xy
r
u
99
defines a cover not by the smooth scheme
singular surface.
by setting
6
=
X , but by a
However, the cover can be normalized
u/v
NO = x
so that
,
and the result is
easily seen to be smooth.
N = 2n
If
2r > n , then the common factors in
and
the higher-order terms of (2.7) and (2.8) are lost and
the action has the form
CU = u + x
(2.9)
av =
v
+ y
where the cover now ramifies only at tne closed point.
Again
y
=
Nv , but now
Nu
=
xy 2 r-n , and the cover
defined by this equation is smooth only if
2r > n
For
a direct calculation shows that the cover is by
A4(2r-n)-l
is
2r = n .
which, with a suitable choice of coordinates,
given by the equation
uw + v
+ (higher degree terms in v) = 0
The action (2.9) extends to an action on the ambient space
Spec k[[u,v,w]]
defined by
av = v + y
(2.10)
where
ou =
w
Ow
u
=
y = Nv = v(v+yn-r) .
Now
A4(2r-n)-l
purely inseparable extensions, a quotient of
is, up to
X
by the
100
cyclic group whose order
2r-n
m
is the greatest divisor of
relatively prime to 2. Therefore the
Dr
2ns singularity
is a quotient by the dihedral group of order
the group has order
2m
2m , since
and has no quotient of order prime
to 2.
The situation for
to those for even
N .
N = 2n+l
splits into cases similar
The analogies to the basic form
(2.5) are more easily seen if the equation (2.6) is replaced
by the equivalent form
z2 + xyn-rz + x2y + y2r+l = 0 ,
(2.11)
x,-~- x + y
which results from the substitution
substitution is valid only if
r > 1.
r
.
This
r = 0, the
If
equation has the equivalent form
z2 + ynz + x 2 y
=
0
and this singularity has a purely inseparable cover by
X
defined by
y = u2
and
z = v2
.
We remark that this analysis also works for the singularity
D2n
n,
which is equivalently defined by
2
2
n
z + x y + xy = 0
and has a purely inseparable cover by
y = u2
and
x = v
101
We will see below that such purely inseparable covers exist
r = 0
for all
If
larity
forms.
n > 2r
Dr
2n+l
in the equation (2.11), then the singu-
is obtained as a quotient by the action
au = U + xyn-2r
av = V + yn-r
where both
even
Nu
and
Nv
equal
y
.
As in the case of
N , this cover is ramified in dimension 1, but the
y = Nu
equations
y = Nv
and
now define a cover by
A1 ,
which, with a suitable choice of coordinates, has the form
uw + v2 + (higher degree terms in
The action on
(2.10).
Since
k[[u,v,w]]
A1
is again given by the equations
has a purely inseparable cover by
X ,
D n+1 (n > 2r) is, up to purely inseparable
the singularity
extensions,
v) = 0
a quotient by the ramified
In the case
2Z/2-action
a
alone.
n < 2r , the covering action is defined
by
au = u + x
av = v + yn-r
Nv = y
where
and
Nu
=
y4 r2n+l , and this action is
unramified in dimension > 0.
of type
A2(4r-2n+l)-l
,
The cover is a singularity
which has equation
uw + v8r-4n+2 + (higher degree terms in v) = 0
102
Again the induced action of
by (2.10).
a
Here, however, the
on
k[[u,v,w]]
A
singularity is the
4
is given
quotient of
X
that
is a quotient by the dihedral group of order
Drn+1
2(4 r-2n+l)
by a cyclic group of order
so
n < 2r
for
The
r-2n+l
singularities are defined in characteristic 2
E6
by the equations
E60: z2 + x3 + y2z = 0
1
2
3
2
E6 : z + x + y z + xyz = 0
(2.12)
In characteristic 0, the
of
X
E6
singularity is
by the binary tetrahedral group
7 .
the quotient
This group
has a cyclic quotient of order 3 corresponding to the normal
subgroup
P2 , and so the singularity has a degree 3 cover
by the double point
D4
Because the degree is prime to
.
the characteristic, this cover carries over to the
characteristic 2 singularities.
The substitution
z = t3
(2.13)
defines a 3-cyclic cover of
If
Er
by the corresponding
Dr
r = 1, the substitution (2.13) results, after
normalization, in an equation of the form
+ xyz + x 3 + y3 = 0
z
for new choices of
form to the
D1
x, y, and z, and this is an equivalent
equation given in (2.6).
The action on
103
X
producing this quotient is defined by
(2.14)
x = Nu
where
and
au
=
u + y
av
=
v + x
y = Nv .
This
2Z/2-action commutes
with the tame 2Z/3-action introduced by the cover (2.13).
T
Let
where
denote this
2Z/3-action, defined by
TU
=
CU
TV
=
C~V
,
C is a cube root of unity.
Then the
is obtained as a quotient of the smooth scheme
cyclic group
G
of order 6 generated by
The situation for
unramified
E6
2Z/2-action of
by the ramified
a
singularity
E6
and
X
by the
T .
is analogous, but here the
a
producing
2Z/2-action for
D4
.
D1
is replaced
(Alternatively, one
could replace it
by the purely inseparable extension which
also gives a cover X -+D40 .)
In both the r = 0 and
r = 1
is
cases, the quotient of
the tame
2Z/3-quotient
X
A2
by the automorphism
We therefore obtain the
diagram of covers
T
(2.15)
A2
T
Dr
/
E6
104
where the covers marked by
those marked by
a
if
are tame of degree 3 and
are degree 2 and are unramified in
r = 1
dimension > 0 if
T
and ramified (or purely inseparable)
r = 0 .
E7
The singularities of type
are defined by the
equations
(2.16)
E :
E1 :
2
E2:
z2 + x 3 + xy3 = 0
2
0
+ x 3 + xy 3 + xyz=
2
z
7. z
+x
+xy
yz =0
E: z2 + x 3 + xy 3 + xyz
In character 0, the
E7
=
0
singularity is the quotient of
by the binary octahedral group
0
whose only quotients
have order divisible by 2; hence the cnaracteristic 2
actions are completely wild.
0
The singularity E
has a purely inseparable cover
by
X
of degree 2, defined by
x =u 2
The singularity
E7
and
y = v2
.
has a ramified double cover with
action of type (2.4), defined- by
au = u + xy
av = v + x
where
x = Nu
and
xy=N.
T
the equations
Though t
X
105
Nu = x = u2 + xyu
Nv = xy = v2 + x2v
(2.17)
do not define a smooth covering for
r
by setting
v/u
=
then
,
Nv = y
E7
,
if we normalize
and the resulting
cover is smooth.
The equation for
Nu
ramified double cover for
given in (2.17) also defines a
E2 . Now, however, normalization
of the resulting cover yields a surface with singularity
D0
,
which itself has, a purely inseparable cover by
53
It is the quotient of
above.
was studied in §1 (ii)
E
Finally, the singularity
7
X
X
by a cyclic group of
order 4 (see (1.12)) and has as an intermediate
Z/2-quotient
D1
the rational double point
The last family of rational double points are those
of type
E8
,
which are defined by the equations
E0: z22 + x3 + y5 =
E : z2 + x3 + y 5 + xy3z
E82: z2 + x 3 +y
(2.18)
3
. z 25+ x
EB8:
+ xy 2 z = 0
+ y5 + y 33z = 0
E : z 2 + x3 + y5 + xyz
As usual, the
cover by
E
E
=
.
singularity has a purely inseparable
X defined by
2
x =u
Both
= 0
and
E
and
y = v2 .
have double covers given by actions of
106
The action producing
the type (2.4).
au =
u
+ y3
av =
v
+ xy
E
has the form
(2.19)
where
by
X
X
x = Nu
y = Nv
and
This defines a double cover
.
ramified in dimension > 0.
For
2
E8
,
the cover by
given by
ramifies only at the origin and the action is
au = U + y
av = v + x
x = Nu
where again
and
x = u2 + y3 u
The equation
Nu
in
y = Nv.
from the definition of
(2.19) also defines a ramified double cover for the
EA
singularity
.
Normalization of the cover yields a
surface with double point
E2
(This could, alternatively,
.
have been realized as a purely inseparable cover defined by
x = u2.)
However, we have not yet been able to compute the
action for which this is the quotient-map.
The most complicated of the
This double point is
4
E8
singularities is
obtained as a quotient of
X by the
metacyclic group of order 12, which is generated by two
elements,
a
and
T , satisfying the relations
a4 = 1,
The tame
2Z/3-automorphism
3 = 1 , and. a'r= 2a.
T
acts on
X
by
E8.
107
Tu
=Cu
1~A ,
Tv =
where
C is a cube root of unity, and the wild 2Z/ 4 -action
is defined by
a
au = V
av = u + g
ag = N 2 u = u(u+g) . (These
2 v = V(v+ag) and
a
a
definitions are-not circular; they may be solved recursively
where
g = N
for both
g
in terms of
ag
and
that the action of
u
and
v .)
We note
can be expressed in the more conven-
a
tional 2Z/4-form (1.12) by introducing the new coordinates
u
=
v = v , and setting
u + v + g ,
a2
Now the action of
is precisely the 2Z/2-action
(2.14) producing the double point
elements
a
and
whose quotient is
T
f = ag .
D4 .
Moreover, the
together generate the group action
E6 .Consequently,
11
the
can be realized as the degree 2 quotient of
Eg8
E6
singularity
corre-
sponding to the quotient of the metacyclic group by the
subgroup generated by
(ii)
a2
and
T
Actions in characteristic 3.
In characteristic 3, the rational double points of
type
An
and
Dn
occur only in their characteristic zero
forms and, up to purely inseparable extensions, the actions
producing them are the maximal tame quotients of the
108
We therefore begin our analysis
characteristic zero groups.
of the characteristic 3 singularities by examining the
E6
forms.
There are two
E6
singularities in characteristic 3:
E60: z2 + x3 + y4 = 0
4 22
3
1
2
E6 : z + x + y +x y
(2.20)
The first is the classical form for
The singularity
E1
B6
and it has a purely
E6
X , defined by
inseparable degree 3 cover by
z = u3
= 0
and
y = v3
.
has a separable degree 3 cover by
which is unramified in dimension > 0.
1 i
E6 is given by
X
The action producing
au = u + y
(2.21)
av = v + u
where
Nv = y
and
Nu = z
.
Note that this is the basic
maximal block form for an unramified
2Z/3-action discussed
in Chapter III (see, for example, Theorem (111.3.12) and
Corollary (111.4.15)).
The
(2.22ce
(22)1
E
singularities are defined by the equations
7:
E:
z2 +x 3 +xy 3 = 0
2
3
3
22
z + x + xy + xy = 0
.
The characteristic zero singularity has a degree 2 cover
109
E6 , corresponding to the index 2 subgroup
by
7
in
0.
This cover is preserved in the characteristic 3 forms, the
substitution
x = t2
(2.23)
defining a tame
C) double cover of
by the corresponding
Er
7
Er
E6'
The action on
X
which produces the double point
is generated by the tame automorphism T of order 2
E
introduced by (2.23),
Tu
=
-u
TV
=
-v
and by the wild 2Z/3-automorphism
As in the case of the covers
a
Dr -.
4 Er
6
defined in (2.21).
in characteristic 2,
the wild and tame actions commute, giving rise to the
diagram of covers
E6
ECA
7
If
r = 0, the separable degree 3 covers become purely
inseparable,
cover of
D
-
Er
E06
resulting
00 in
by
X
.
the cover of
E0
by
A
and the
Again note the analogy with the
covers in characteristic 2.
110
The
singularities in characteristic 3 have three
E8
possible forms:
(2.24)
E :
z2 + x3 + y5 =0
E1:
z2 + x3 + y5 + x 23 _
E8:
z2 + x 3 + y5 +x
The classical form
cover by
X
0
E
2
y2 =
has a purely inseparable
of degree 3 defined by
z =u3
and
y = v.
The substitution
t3
x2 + y2
defines a purely inseparable degree 3 cover of
E81
by
E0
6 '
whicn itself has a purely inseparable cover by X
2
More interesting is the case of E8 . If the basic
2Z/3 -action (2.21) producing
Nv = y 2
instead of
is modified by setting
y , we obtain a 3-cyclic cover of
ramified only at the origin.
Nu =
E6
E2
The resulting equations
z = u(u+y)(u+2y)
Nv = y 2 = v(v+u) (v+2u+y)
D4 , which in charac-
define a cover by the double point
teristic 3 is a tame quotient of
X
by the quaternion group
therefore an
2 . The action on X producing E 8 is
extension of the cyclic group of order 3 by the group a2
111
and, since it has no quotient prime to 3,
the binary tetrahedral group
(iii)
it is given by
7 .
Actions in characteristic 5.
singularities have
E8
In characteristic 5 only the
5 ; the other singularities
no covers of order prime to
exist only in their classical forms.
The
E8
singularities
are defined by the two equations
(2.25)
E0: z2 + x3 + y5
2
1
E8 : z + x + y + xy 4 = 0,
by
X
has a purely inseparable cover
E8
The double point
of degree 5 defined by
z = u5
and
Artin [ 3 ] has shown tiat the
5-cyclic cover by
E8
x = v5
form has a separable
defined by
X
u
However, the action on
-
X
auotient still eludes us.
y4 u + xz = 0
which produces
E8
as its
112
REFERENCES
1. Altman, A. and Kleiman, S. L., "On the Purity of the
Branch Locus", Compos. Math. 23 (1971),461-465.
2.
Artin, M., "Wildly Ramified 2Z/2 Actions in Dimension
Two", Proc. Amer. Math. Soc. 52 (1975), 60-64.
3.
Artin, M., "Coverings of the Rational Double Points in
Characteristic p ", in Complex Analysis and Algebraic
Geometry, Cambridge Univ. Press, Cambridge, 197/.
4.
Brieskorn, E., "Rationale Singularituten Komplexer
FlUchen", Invent. Math. 4 (1968), 336-358.
5.
Cartin, H., "Quotient d'un Espace Analytique par un
Groupe d'Automorphismes", in Algebraic Geometry and
Topology, Princeton Univ. Press, Princeton, 1957.
6.
Ellingsrud, G. and Skjelbred, T., "Profondeur d'Anneaux
d'Invariants en Caracteristique p ", Comptes Rendus,
286, ser. A (1978), 321-322.
7.
Fogarty, J., "Local Invariants of Finite Groups",
(preprint).
8.
Fossum, R. M. and Griffith, P. A., "Complete Local
Factorial Rings which are not Cohen-Macaulay in
Characteristic p ", Ann. Scient. Ec. Norm. Sup. 8
(1975), 189-199.
9.
Grothendieck, A., Revetements Etales et GrouDe
Fondamental (SGA 1), S4minarire de Gsomstrie Algebrique
du Bois Marie 1960/61, Lecture Notes in Math. 224,
Springer-Verlag, Berlin and New York, 1971.
10.
Grothendieck, A., Local Cohomology (notes by
R. Hartshorne), Lecture Notes in Math. 41, SpringerVerlag, Berlin and New York, 1967.
11.
MacMahon, P. A., Combinatory Analysis, Chelsea Publ.
Co., New York, 1960.
12.
Matsumura, H., Commutative Algebra, W. A. Benjamin,
Inc., New York, 1970.
13.
Samuel, P., "Classes de Diviseurs et Dsrivses
Logarithmiques", Topology, 3 (1964), 81-96.
113
14.
Serre, J.-P., Alg'ebre Locale-Multiplicitss, Lecture
Notes in Math. 11, Springer-Verlag, Berlin and New
York, 196515.
Zariski, o., "On the Purity of the Branch Locus of
Algebraic Functions", Proc. Nat. Acad. Sci. U.S.A.
44 (1958), 791-796.