PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 52, October 1975
WILDLYRAMIFIED Z/2 ACTIONS IN DIMENSIONTWO
M. ARTIN1
ABSTRACT.
The
automorphism
Let
series
k be a field
ring over
variants
of power
series
2 are described
which
of characteristic
two,
except
at the closed
the maximal
point.
ideal
that
This
which
that
k[[u, v]\
an
form.
be a power
the ring
R oi in-
under a ^-automorphism
of a on Spec
there
is ff-invariant,
under
a standard
is to study
a, i.e.,
the action
means
invariant
having
and let
Our object
of k[[u, vf] under an involution
2. We assume
are
by equations
k in two variables.
k[[u, v]] oi order
than
rings
of order
is no prime
and such
that
o oi
k[[u, v]] is free
ideal
p. other
the induced
action
on k[[u, v]]/)f> is trivial.
It is known
sumption
cept
that
on fixed
at the closed
we are in effect
mental
k[[u,
v\]
that
point.
It follows
studying
covering
with residue
field
k[[u,
R 111, and in view
is etale
that
and of degree
R is a complete
local
spectrum
is the pointed
over
vll
a complete
group of its pointed
universal
is finite
points,
^-algebra
local
R such
X = Spec R — 'mRS is
spectrum
U oi a regular
of our as-
2 over
ring.
that
Thus
the funda-
Z/2,
local
R ex-
and that its
/e-algebra
k.
Here is the result:
Theorem.
The ring
R can
be defined
in k[[x,
;
,2
y, z]]
by one equation
of
the form
z
where
a, b e k\fx,
any such
cover
equation
k[[u,
v]]
y]]
u
a ring
n
which
are relatively
R having
the above
prime.
Conversely,
properties.
Its double
by the equations
+ au + x = 0,
the action
be interesting
v
of a by a bar,
uu = x,
It would
2
+ abz + a y + b x=U,
are nonunits
defines
is given
and ij we denote
2
vv = y,
to have
+ bv + y = 0,
then
uv + uv = z.
an extension
of this
result
to
Z/p-actions
for p > 2.
Received by the editors July 5, 1974.
AMS (MOS) subject classifications
(1970).
Primary 13B10, 14E20; Secondary
14B05, 14J15.
Key words
and phrases.
Wild ramification,
etale
coverings,
group
actions.
1 Supported by NSF Grant PNDGP-42656.
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Copyright IB 1975. American Mathematical Society
60
WILDLY RAMIFIED Z/2 ACTIONS IN DIMENSION TWO
We consider
o as a pair of power
be given
by a matrix
variable,
the matrix
whose
square
series
in u, v. The linear
is the identity.
will be of the form
61
After
! 0 j t. (It turns
terms
a linear
out that
will
change
of
in fact
f = 0.)
This means
i
u = u + (degree
> 2),
v = v + (u + (degree
> 2).
Set
x = uu = u2 + (degree
Then we obviously
Lemma
have
> 3),
y = vv = v
+ euv + (degree
> 3).
k[[u, v]] 3 R 3 k[[x, y]].
1. k[[u, v]] and
R are free
k[[x, y]]-algebras,
of ranks
A and 2
respectively.
Proof.
hence
It is clear
that
osition
x, y form a system
k[[u, v]] is a finite
22],
clear—a
that
Thus
basis
generically
we need
consists
etale
of parameters
k[[x, y]]-module.
only check
that
of the residues
and of degree
v]]
It is free by [4, IV-37,
dim,
k[[u, v]]/(x,
of 1, u, v, uv.
2 over
in k[[u,
Prop-
y) = A. That
Since
R, R is of rank
and
is
k[[u, v]] is
2 over k[[x, y]]. Again,
it is free by [A, loc. cit.].
Corollary.
Lemma
The multiplicity
2. The field
Proof.
Let
k((u, v))/K
separable
extension
able
k[[x,
over
codimension
Since
y)).
and such
1 (purity
v)) over
of fractions
and unramified
of k((x,
y]],
k((u,
For,
a ring
of R- Then
in codimension
otherwise
cannot
of the branch
k((x,
is Galois.
the field
extension
1 on R. Also,
R would
have
locus
y))
be purely
any extension
K is a
insepar-
unramified
[5], and [2, p. 240, Theorem
in
4.10]).
[K: MOc, y))] = 2, K is Galois over &((*, y)).
Let
S = R ® R and
over the ring
Above
etale
extension
K be the field
is separable
of R is two.
k[[x, y]].
any codimension
and of degree
S —> T.
Since
dimension
1 prime
R (say
and so it is certainly
completely
of k[[x, y]], the extension
S ~
unramified
v]], both tensor
the normalization
S —> T is etale
with
as
v]] ® k[[u,
S, T denote
2. Hence
K is Galois,
1 over
[A], T splits
T = k[[u,
Let
of degree
R x R. Therefore
R acting
over
Proof.
Otherwise,
over
it must
Therefore
be a cyclic
group.
k[[x, y]] at some codimension
of k[[u,
v]] lying
over
p, and let
k(iu,
in co-
product),
1. By purity
v)) is Galois.
y)) is G = Z/2
© Z/2.
We know by purity
that
1 prime
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<? be a prime
and so is
T is unramified
k[[u, v]] in codimension
k[[u, v]]-algebra.
being
rings.
R —> k[[u, v]] is
4 there,
on the left in the tensor
Lemma 3. The Galois group of kiiu, v))/kiix,
is ramified
products
of these
p of k[[x, y]].
H C G be the inertial
R
Let
subgroup
62
of
M. ARTIN
Q, Then
a prime
there
since
<?. of
would
k[[u, v]\
a cannot
k[[u, v]]
be only
By Lemma
and
L' respectively.
such
algebra
These
with
The sets
The
A is ramified.
the inertial
Let
Then
to be B. Thus
Proof.
in common,
natural
rings
map
free,
G were
between
H leads
(and hence
k[[u,
k((x, y)) and
of rank 2. Any
+ bt + rj)
\b - Oi are the ramification
Since
prime nonunits
A, B are ramified,
to an intermediate
not contain
over
vf\ = A ® B, the tensor
the ramification
loci
\), and hence
ring
ct and
product
of A and
k[[u,
v]l
A ® B = k[[u,
in k[[x, y, s, t]], and we apply
mally smooth over k and
but
b are relatively
prime.
being
y]].
over
only
k[[x,
the closed
easily
in codimension
v]\
as the ring
point
that
the
1. Both
by the equations
t2 + bt + 7] = 0
the jacobian
a, b are nonunits,
defined
criterion.
it follows
/dt;/dx
d&dy\
\drj/dx
drj/dyj
Since
that
k[[u, vll
is for-
the jacobian
matrix
is invertible,
hence
that
£, f] is a regular
system
of parameters
This
in turn that
s, t is a regular
system
of parameters
the automorphism
we have
make the change
is un-
and so <7Jis an isomorphism.
s2 + as + £ = 0,
By construction,
lemma,
no choice
1. It follows
A ® B —> k[[u, v]] is an isomorphism
which
which
ring has
B have
in codimension
a, b
so is R.
as in the previous
\3. This
b, and so
Hence
of k[[x, y]] above
above
\s. Then
in k[[x, y]].
by purity.
1 prime
over
all) primes
A ® B is nonsingular
We now view
a is given
s~ = s + a, T = t + b, and
of variable
by the actions
sF = q, tt
u
+ au + x = 0,
v + bv + y = 0.
in kux,
y\\.
in k\\u,
v]].
on each factor of
= rj. So, we can
(x, y, u, v) —►(f, rj, s, t) to obtain
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(*)
have
of k[[x, y]] in L
B = k[[x, y]][t]/(t2
k[[u, v]] is also ramified
are free modules,
A ® B, i.e.,
cyclic,
so we would
k[[x, y]]-algebras
J3 be a codimension
*p does
5.
implies
is
So we may write
!a = 0! and
of k[[u, v\] lying
subgroup
at some
Lemma
L, L
a, b are relatively
k[[x, y]]-algebras
Q be a prime
ramified
2. There
B respectively.
are not invertible.
Let
)f>. If
2, and
the normalizations
are again
Lemma 4. The elements
Proof.
two fields
+ as + £),
a, b, t;, tj e k[[x, y]].
over
order
of \).
by one element.
A = k[[x, y]][s]/(s2
loci of A and
the choice
A, B denote
is generated
R, H has
H oi order
are exactly
K. Let
over
is unramified
subgroup
contradicts
3, there
k((u, v)) besides
which
one
= R. This
be ramified
equations
WILDLY RAMIFIED Z/2 ACTIONS IN DIMENSION TWO
Conversely,
sider
let
a, b £ k[[x, y]] be any relatively
the extension
given
theory that they define
over, the jacobian
issmooth,
where
by the equations
a Galois
criterion
and equal
with group
ffi Z/2.
by these
More-
equations
Let A, B, R be the three intermediate
+ au + x), and
Lemma 6. With the above
and conby Galois
G = Z/2
that the ring defined
to k[[u, v]].
A = k[[x, y]][u]/iu2
prime nonunits,
(*). It is immediate
extension
shows
63
notation,
B = k[[x, y]][v]/iv2
rings,
+ bv + y).
k[[u, v]] is unramified
in codimension
1 over R.
Proof.
etale
Clearly,
over
k[[u, v]] is the normalization
k[[x, y]] at each point
A ® R, and hence
k[[u,
of U = Spec k[[x, y]][l/a],
v]], is e'tale
Similarly,
k[[u, v]] is etale
relatively
prime,
of A ® R. Since
over
over
R except
R except
above
A is
it is clear
the locus
above
\b = Ot. Since
R. Let
z = uv + uv,
that
\a = Ot.
a and
b are
the lemma follows.
We now ask for the equation
u + a and v = v + b. Clearly
tion for z over
defining
z £ R, and z = ub + va.
k[[x, y]] is easily
seen
where
u =
The irreducible
equa-
to be
/ = z + abz + a y + b x = 0.
Therefore
that
this
k[[x, y, z]]/if)
equation
codimension
that
is birationally
defines
1. This
a = 0, hence
a normal
is clear
equivalent
ring,
except
i.e.,
to R. It remains
to verify
that the ring is nonsingular
on the ramification
locus
in
\ab = OS. Say
b / 0. At such a point,
df/dx = ida/dx)bz+
b2.
Let a = da/dx.
Then if df/dx = 0, it follows that a'z + b = 0. Substitution
of this equality
into
but x is not a unit,
pletes
the proof
classified.
and hence
can choose
cannot
Let us assume
1 are easily
be normal,
this
to a'
x = 1. Since
a'
is an integral
power
hold anywhere
on Spec
k[[x, y, z]].
k algebraically
closed.
Involutions
This
series
com-
of the Theorem.
Examples.
sion
/ leads
If a acts
a power
a generator
series
on &[[a]], then the invariant
ring
z for the field
^[[i*]]. By Artin-Schreier
extension
such
in dimenring
will
theory,
that
0
z2 - z = cp = Z
a/
i=-N
and such that only odd negative
Write
d> = ut~
where
v T~
r+1, where
= u results
indices
zz is a unit.
a. occur
Then
in the expression
a change
in an equation
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z2-z
= r2'
+ 1.
of variable
for cb.
t' - tv,
we
64
M. ARTIN
The element
s = tTz is a local
parameter
for /e[[zz]], satisfying
the equation
s2 + fs + t = 0.
So, this
is the normal
volution
corresponding
form in dimension
to this
1. (The only case
equation
is rational
is
in which
r = 1, where
the in-
it is the
action
S ss-> s/(l
We can obtain
on the variables
examples
+ s) = s + s
+ s
in dimension
u, v by some
+•••.)
2 by letting
of the above
actions.
o act independently
This
leads
to the cases
a = x1, b = y1:
u2 + x'u + x = 0,
This
say
equation
defines
v2 + y'v + y = 0;
a rational
i = 1, it is a double
z2 + x'y'z
singularity
point
of type
D
+ x ly + xy 7 = 0.
[3] if and only if z or / = 1. If
with
ra = 4j:
a
z 2 + xy i z + x 2 y + xy 2V
' = U.
Setting
a = y, b = x
leads
to a rational
double
point
of type
Eg
[3, p.
270]:
r,
z 2 + x 2 yz + y 3 + x 5 = 0.
REFERENCES
1. A. Grothendieck,
Techniques
geometrie
algebrique.
Ill, Seminaire
fasc.
1, Secretariat
mathematique,
2. A. Grothendieck
Seminaire
de Geometrie
J. Lipman,
Paris,
et theoremes
d'existence
anne'e:
1960/61,
Expose
Berlin
Rational
and unique factorization,
en
212,
1961. MR 27 #1339.
(editor),
Revetements
etales
et groupe fondamental,
Algebrique
du Bois Marie 1960/61,
Lecture
Notes
vol. 224, Springer-Verlag,
3.
de- construction
Bourbaki
13ieme
(SGA 1),
in Math.,
and New York, 1971.
singularities,
Inst. Hautes
with
Etudes
applications
to algebraic
surfaces
Sci. Publ. Math. No. 36 (1969),
195—
279- MR 43 #1986.
4.
J.-P.
Serre,
Algebre
no. 11, Springer-Verlag,
5. O. Zariski,
locale.
Multiplicites,
2ieme
ed.,
Lecture
Notes
in Math.,
Berlin and New York, 1965. MR 34 #1352.
On the purity
of the branch
locus
of algebraic
functions,
Proc.
Nat. Acad. Sci. U. S. A. 44 (1958), 791-796. MR 20 #2344.
DEPARTMENT OF MATHEMATICS, MASSACHUSETTS INSTITUTE OF TECHNOLOGY,
CAMBRIDGE, MASSACHUSETTS 02139
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