C OMPOSITIO M ATHEMATICA
J EFFREY L ANG
The factoriality of Zariski rings
Compositio Mathematica, tome 63, n
o3 (1987), p. 273-290
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273
The factoriality of Zariski rings
JEFFREY LANG
Mathematics Department, University of San Francisco, San Francisco, CA 94117, USA Present address: Mathematics Dept., University of Kansas, Lawrence, KS 66044, USA Received 10 November 1986; accepted 19 February 1987
Compositio (1987)
Martinus Nijhoff Publishers, Dordrecht - Printed in the Netherlands
Introduction
Let k be an
algebraically
closed field of characteristicp ~ 0,
g Ek[x, y]
besuch that gx
and gy
have no common factors ink[x, y],
E cAk
be the surfacedefined
by
theequation
zP =g(x, y)
and A =k[xP, yP, g].
Inprevious
articles
(see [1], [3] and [ 13])
E was called a Zariski surface andattempts
were made to findgeneric
conditions on g that would force the coordinatering
of E to be factorial. These papers used the fact that the coordinate
ring
ofE is
isomorphic
to A and somepartial
results were obtained.In this article the divisor class group of these surfaces is
investigated
froma
slightly
differentangle.
Let Fbe anon-algebraically
closed field of charac- teristicp ~
0. LetF
be analgebraic
closure of F. Given g inF[x, y]
letFg
be the field extension of F obtained
by adjoining
the coefficientsof g
to F.This paper
investigates
therelationship
between thesingular points
of thesurface zp =
g(x, y)
ink2
and the divisor class group of thering Fg[xp,yp, g].
After some
preliminary
results in Section1,
Zariskirings
are discussed inSection 2. In this section
singularity
conditionsaffecting
the order of the divisor class group of a Zariskiring
arepresented.
Some
general
facts about Zariskirings
appear in Section 3.In Section
4,
the main section of thearticle,
the fact that for p >3,
Zariski
rings
are factorial for ageneric
choiceof g
isproved by showing
thatfor a
generic
g, the class group of the surface zp = g is trivial.Section 5 closes this article with a theorem about
logarithmic
derivatives of the Jacobian derivation and some openproblems.
0. Notation
(0.1) GF(pn) -
the finite fieldwith pn
elements.(0.2)
F - a field ofcharacteristic p ~
0.(0.3) F -
analgebraic
closure of F.(0.4)
For g EF[x, y]
we denoteby Fg
the field extension of F obtainedby adjoining
to F the coefficientsof g.
(0.5) For g
EF[x, y]
we denoteby Ag
thering Fg[xp, yP, g].
We call theserings
Zariskirings.
(0.6)
If A is a Krullring
we denoteby Cl(A)
the divisor class group of A.(0.7) Surface-irreducible, reduced,
two dimensionalquasiprojective
var-iety
over analgebraically
closed field.(0.8)
If E is a surface we denoteby Cl(E)
the divisor class group of the coordinatering
of E.(0.9)
k - analgebraically
closed field of characteristicp ~
0.(0.10) Ak -
affine n-space over k.(0.11)
Kn - the set of alln-tuples
of elements of k.(0.12) For g
Ek[x, y]
we letSg = {(03B1, f3)
Ek2 : gx(03B1, 03B2)
=gy(03B1, 03B2)
=01.
1. Preliminaries
The
following results, (1.1)
to(1.4),
can be found in P. Samuel’s 1964 Tata notes[17].
For the definition of a Krullring
the reader is referred to either Samuel’s notes or R. Fossum’sbook,
"The Divisor ClassGroup
of a KrullDomain"
[5].
All of therings
considered in this paper are noetherianintegrally
closed domains and are therefore Krullrings.
THroRmvt 1.1. Let A c B be Krull
rings. If
eachheight
oneprime of
Bcontracts to a
prime of height
less than orequal
to oneof
A then there isa well
defined
grouphomomorphism ~: Cl(A) ~ Cl (B). If
B isintegral
over A or
if B is A-flat
then this condition issatisfied. (See [17]
pp. 19-20for details.)
REMARK 1.2. Let B be a Krull
ring
of characteristicp ~
0. Let A be a derivation of thequotient
field of B such thatA(B)
c B. Let K = ker0394 and A = B n K. Then A is a Krullring
with Bintegral
over A. Thusby (1.1)
there is a well-defined
map ~: Cl(A) - Cl(B).
SetL - {t-1 0394t : t belongs
tothe
quotient
field of B andt-1 0394t
EBi
and L’ ={u-1 0394u :
u is a unit inB}.
Then L’ is a
subgroup
of L.THEOREM 1.3.
(a)
There exists a canonicalhomomorphism : ker ~
~L’/L’.
(b) If L is
thequotient.field of
B and[L: K]
= p andA(B)
is not containedin any
height
oneprime of B, then ~
is anisomorphism ([17]
pp.63-64).
THEOREM 1.4.
If [L: K] =
p, then(a)
there exists an et E A such that AP = 03B10394 and(b)
an element t E K isequal
toDv/v for
some v E Kif
andonly if AP-’t -
at = -tP([17)
pp.63-64.).
REMARK 1.5. These
results, (1.6)
and(1.8)
are to be found in[11]
pages 394-395. These theorems assume that F is a field of characteristicp ~ 0, g(x, y)
EF(x, y]
is such that gx and gy have no common factors inF[x, y].
THEOREM 1.6.
(Ganong’s Formula)
Let D :F(x, y) ~ F(x, y)
be the F deri-vation
defined by
D =gy(~/~x) - gx(iJ/oy). Then for
each et EF(x, y),
where DP = cD and V =
~2P-2/~xP-1~yp-1.
REMARK 1.7. In
[11]
the writerproved
this result for the casedeg (gx) = deg (g) -
1. In[16]
Stôhr and Volochproved
this formula ingeneral.
THEOREM 1.8. Let D =
gy(~/~x) - gx(~/~y).
Let Y be the additive groupof logarithmic
derivativesof
D inF[x, y] (See (1.2).)
and A =F[xP, yP, g].
Then
(i) D-1 (0) n F[x, y] = A, (ii) Cl(A) - L,
(iii) t
E Limplies
thatdeg
tdeg (g) - 2,
(iv)
The coordinatering of
thesurface defined by
zP =g(x, y)
isisomorphic to A Q9 F.
(See [11] pp. 393-394.)
2.
Singularity
conditions on Zariskirings
REMARK 2.1. A surface in affine
3-space
definedby
anequation
of the formzP =
g(x, y)
withonly
a finite number of isolatedsingularities
is called a Zariskisurface,
where theground
field isalgebraically
closed of characteris-tic p ~
0. The coordinatering
of such a surface isisomorphic
tok[x°, yP, g]
where k is the
ground
field([11]
p.393). Hereafter,
in this paper allrings
ofthe form
F[xP, yP, g]
where Fis afield,
notnecessarily algebraically closed,
of characteristic
p ~ 0
will be referred to as Zariskirings.
This section studies Zariskirings
defined overnon-algebraically
closed fields.An
important
tool is thefollowing
lemma.LEMMA 2.2. Let D :
k(x, y) ~ k(x, y)
be the k-derivationdefined by
D =gv(~/~x) - gx(~/~y)
and c be such that DP = cD.If (a, b)
Ek2
is such thatg, (a, b) = gv(a, b) = 0,
thenc(a, b) = (H(a, b»P -’
whereH(x, y) = g2xv -
gxxgyy.Proo, f :
For each a Ek(x, y),
by (1.6).
Set a -
1,
then c -03A3p-1l=0gl~(gp-l-1).
Let g = g(x +
a, y +b)
and c =03A3p-1i=0l~(p-i-1).
Thenc(0, 0) = 03A3p=1l=0g(a, b)l~(gp-l-1) (a, b) = c(a, b). By Taylor’s formula,
Thus
Let = g - g(a, b)
and = -03A3p-1i=0~(gp-i-1).
Since(g)x = ()x
and()l = (g)v
it follows thatc(x, y) - c(x, y)
andc(O, 0) = c(a, b).
Since(0, 0) =
0 it follows thatc(O, 0) = ~(p-1)(0, 0).
Asimple
calculationyields
that the lowestdegree
term ingP-l
isThus the lowest
degree
term ofV(gP-l)
is the constant term,In the
previous step
a combinatorialidentity
was used(see [6]
page90, identity z.40).
Thus the constant term inV(gP-l)
is(H(a, b))(p-1)/2.
Therefore~(p-1)(0,0) = ( H(a, b))P - 1 .
REMARK 2.3. Let F be a non-algebraically closed field of characteristic p -A 0
and
F an algebraic
closure of F. For g EF[x, y],
letFg
be the field extension of F obtainedby adjoining
to F the coefficients of g.Throughout
theremainder of this article g will
always satisfy
two conditions(1)
gxand gy
have no common factors inF[x, y]
and that gxand gy
intersectin the maximum
possible
number ofpoints
in2((n - 1)2 if n ~ 0 (mod p), n2 -
3n + 3otherwise,
where n =deg (g).),
and(2)
gx, gy and H= gxy -
gxxgyy are neversimultaneously
zero at anypoint
in
P2 (see [1] for
thegeneric
nature of theseconditions).
The effect ofthese conditions and others on the divisor class group of
Ag = Fg [xP, yP, g]
will beexplored
in the rest of this paper. Theassumption
willalways
be made that g has no monomials of the formxrPysP,
sinceFg[xP, yP, g] = Fg[xP, yP, g
+xrpyspl.
·THEOREM 2.4.
If the
ideal I =(gx, gy)Fg[x, y]
nFg [x]
inFg [x]
isprime
andif
no twopoints of Sg = {(03B1, 03B2)
E2: gx (a, 03B2)
=gy (a, 03B2)
=01
have thesame x-coordinate
then for
each(a, b)
ESg, the field degree [Fg(a): Fg] equals
Proof.
Consider the case n ~ 0(mod p).
Letf(x)
be the resultant with respect to xof gx
andgy .
Thenf (x)
is ofdegree (n - 1)2
andbelongs
toI([15]
page186).
I is aprincipal
idealgenerated by
apolynomial
ofdegree
at least
(n - 1)2 .
Therefore I =(f(x)). If (a, b) E Sg then f(a) =
0 whichimplies
that[Fg (a) : Fg ]
=(n - 1)2 .
The n --- 0(mod p)
case is similar.COROLLARY 2.5.
If m
=(gx’ gy)Fg[x, y]
is aprime
ideal inFg[x, y]
andf
notwo
points of Sg
have the same x-coordinate or the samey-coordinate,
thenFg(a, b) = Fg(a) = Fg (b), for
all(a, b)
ESg.
Proof. By (2.4)
both a and b areseparable
overFg
ofdegree equal
to thenumber of elements in
Sg.
ThenFg (a, b)
isseparable
overFg
ofdegree equal
to the number of
Fg -injections of Fg (a, b)
into([15],
p.65).
Since each suchinjection
must take an elementof Sg into
another elementof Sg it
follows that[Fg(a, b) : Fg(a)] = [Fg(a, b) : Fg(b)] =
1.COROLLARY 2.7.
If no
twopoints of Sg
have the same x or y coordinate and bothof
the ideals(gx, gy)Fg[x, y]
nFg[x]
and(gx, gy)Fg[x, y]
nFg[y]
areprime
thenFg(a)
=Fg (b)
=Fg (a, b).
REMARK 2.8. Let k be an
algebraically
closed fieldof characteristic p ~ 0
and D:
k[x, y] - k[x, y]
be definedby
D =gy (ôlôx) - gx(~/~y) .
Let Y bethe group of
logarithmic
derivatives of D ink[x, y]. By (1.4)
an elementt E
k[x, y]
is in L if andonly
ifDP-1 t -
ct = - tP where DP = cD. Itfollows that if
(a, b)
ESg,
thenc(a, b)t(a, b)
=t(a, b)P,
whichby (2.2) implies
that(t(a, b))p
=(H(a, b))P-1(t(a, b)).
SinceH(a, b) ~
0by
con-dition
(2),
the set of solutions in k to thepolynomial equation
zP -( H(a, b))p-1 z
= 0 isisomorphic
toZ/pZ.
Thus 0 : Ef -Z/pZ
definedby 0(t)
=t(a, b)H(a, b)
is ahomomorphism
of additive groups.THEOREM 2.9. Let g
satisfy
conditions(1)
and(2). If
0 ~ t E Y thent(Q) ~
0for
at leastpoint.s Q
ESx,
where n =deg (g).
Proof.
Let 0 ~ t E L.By
condition(1),
each irreducible factor of t ink[x, y]
is
relatively prime
to either gx or gy. Therefore t can be factored ink[x, y]
as t = uv where u is
relatively prime
to gx and v isrelatively prime
to gy(If
t is
already prime
to gx then let u = t and v =1.).
Then umeets gx
in atmost
(n - 1) deg(u) points
and vmeets gy
in at most(n - 1) deg(v) points.
Thus u
(resp. v)
is 0 at most(n - 1) deg(u) (resp. (n - 1) deg(v)) points
ofSg.
Thisimplies
that t is not 0 for at leastpoints
of S. Sincedeg(u)
+deg(v)
=deg(t)
the desired result is obtained.COROLLARY 2.10 Let g
satisfy (1)
and(2). If 0 ~
t E Y thent(Q) ~ 0 for
at least
(n - 1) points of Sg if n e
0((mod p)) and for
at least onepoint of
Sg
otherwise.Proof. By (1.8) deg t n -
2. The result is now an immediate consequence of(2.9).
COROLLARY 2.11. Let g
satisfy (1)
and(2).
Then thehomomorphism
03A6: L ~
~Q~Sg Z/pZ 2022 H(Q) defined by 03A6(t) = (t(Q))Q~Sg
is aninjection.
COROLLARY 2.12.
If (gx, gy)Fg[x, y]
nFg[x] is prime
inFg[x]
andif
no twopoints of Sg
have the same x-coordinate then the restrictionof
0: L ~Z/pZ
to
Lg = L
nFg [x, y] is
aninjection.
Proof.
For t E!£g, 0(t) = t(a, b)
where(a, b)
ESg. Suppose
that0(t) =
0.Let
(a’, b’)
ESg.
As in theproof
of(2.5)
there exists anFg -isomorphism
fromFg(a,b) onto Fg(a,b) such that 03C3(a) = a’ and a(b) = b’. Since t(a, b) = 0,
then
a(t(a, b)) = t(a’, b’) =
0. Therefore (D as defined in(2.11)
maps t to 0 in~Q~Ss Z/pZ 2022 H(q) . By (2.11), t
= 0.DEFINITION 2.13. The conditions on g that no two
points
ofSg
have the samex-coordinate and that
(gx, gy)Fg[x, y]
nFg[x]
is aprime
ideal inFg[x]
willhereafter be referred to as conditions
(3)
and(4) respectively.
THEOREM 2.14. Let g
satisfy
conditions(1) - (4).
LetAg = Fg [xp, yP, g]. If
p -
2,
thenCl(Ag) ~ Z/2Z. Ifp
>2,
thenCI(Ag)
is trivial or isisomorphic
to
7L/p7L.
PROOF The p > 2 case is an immediate consequence of
(2.12).
Assume thenthat p = 2. Then D(gx)/gx = (gxxgy - g,gx)lg., =
gxy is a nonzero element ofYg by
condition(2). By (2.12), Cl(Ag) ~ 7L/27L.
EXAMPLE 2.15. If p >
2, g - x2 - y2
and F =GF( p),
then g satisfies conditions(1)-(4).
Since zP= x2 - y2
isclearly
notfactorial, Cl(Ag) ~ Z/pZ.
EXAMPLE 2.16 Let k be an
algebraically
closed fieldof characteristic p ~
0.Let n
4 be a positive integer. Let {Tij:0 i + j n}
be a set of indeter- minates over k. Let F =k(Tij)
and g =03A30i+jn Tijxiyj.
Then g satisfies conditions
(1)-(3).
To see this letR(x)
be the resultant withrespect
to xof gx
and gy. ThenR(x) ~
0. This can be demonstratedby showing
that for somespecialization
of theTij, R(x) ~
0. If n is not divisibleby p, then g = xy
+(1/n)(xn - V) gives R(x) = x(n-1)2
+ x.Furthermore,
if D is the discriminant ofR(x),
then D is a nonzeropolynomial expression
in theTi,. Again
this can be shownby demonstrating
that D ~ 0 for some
specialization
of theTij.
Forexample, if n ~ 0,
2(mod p),
and g = xy +(1/n) (xn - yn ), then
D =n(n - 2). Similarly,
it iseasy to show that if
R(x)
is the resultantof gx
and H andif D
is the resultant ofR(x)
andR(x),
thenD
is a nonzeropolynomial
in the7Ç, . Again
if wespecialize
andlet g = xy
+(1/n) (xn - yn )
thenD becomes n2 -
2n + 2.One concludes that
(a) R(x)
is a nonzeropolynomial
in theTij
and x ofdegree
in xequal
to(n - 1)2 if n e 0 (mod p), of degree n2 -
3n + 3 otherwise. Therefore gxand gy
arerelatively prime,
(b)
D is a non-zeropolynomial
in theTij
whichimplies
that gx and gy intersect in the maximumpossible
number ofpoints
inF2.
(c) D
is also a nonzeropolynomial
in theTij
whichimplies
condition(2).
(b)
above alsoimplies
condition(3). (See [18]
pages 23 to 31 for further discussion on theresultant.)
REMARK 2.17. Note that for any
specialization
of theTij
for whichR(x), D,
and D
become nonzero, then for that choiceof g
conditions(1), (2)
and(3)
will be met. Thus conditions
(1), (2)
and(3)
aregeneric
conditions on g.(2.16 continued ...)
Condition(4)
is also met. First ofall,
gx = tlO +2t20x
+ t11y +... and gy = t01
+2t02y
+ t11x+ ... Then k[Tij[[x, y]/
(gx, gy) = k[too,
t20, tl 1t02, ...] [x, y]. Therefore gx and gy generate a prime
ideal in
k[tij] [x, y]. By
condition(1),
the idealgenerated by
gx and gy ink[tij] [x, y]
does not meet themultiplicatively
closed setgenerated by
thenonzero elements of
k[Tij]. Thus gx and gy generate
a maximal ideal ink(Tij) [x, y], implying
condition(4).
ThereforeCl(Ag) ~ 7L/27L if p =
2 andCl(Ag) ~
0 or7L/p7L
if p > 2.(For p
5 see(2.34)).
Question
2.18. Is condition(4)
ageneric
condition ong?
THEOREM 2.19. Let g
satisfy
conditions(1) - (3).
Let(f(x)) = (gx, gy)Fg [x, y] ~ Fg[x]. Suppose that f(x) factors
into aproduct of r-irreducible factors
in
Fg[x] .
Then the orderof CL(Ag) p’ .
Proof :
Letf(x) = f1 (x)
...fr(x)
be a factorizationof f(x)
inFg[x]
intoprime
factors. For each i =1,
..., r, let 03B1i be a rootof fi(x) in F.
For eachi,
there isa 03B2i
EF
such that(al , 03B2i)
ESg. Let e: Yg --+ ~ri=0 Z/pZ
be definedby (t) = (t(ai, 03B2i)/H(03B1i,03B2i))ri=1.
Let t ~ker ë
and let(a, 03B2)
ESg.
Thenfi(ot) ==
0 for some i =1,
... , r. Therefore et isconjugate
to oc, so thatthere
exists anFg-automorphism 03C3 : F ~ F
such thata(ai)
= a . Thenu(oci, 03B2i)
=(a, 03B2).
Sincet(a; , PJ
= 0 thisimplies
thatt(a, fi)
=03C3t(03B1i, 03B2i)
= 0.By (2.11)
t is
identically
0.Thus P
is aninjection. By (1.8),
the order ofCl(Ag) p’ .
281 REMARK 2.20. The ideal
generated by f(x)
in(2.19)
is identical to the idealgenerated by
theresultant, R(x), of gx and gy
withrespect
to x. This isbecause in this case,
R(x)
isof degree equal
to the number of elements inSg.
Since
R(x)
E(f(x)) and f(03B1)
= 0 for each(a, 03B2)
ESg,
then(R(x)) = (f(x)).
(See [15]
p.185.).
EXAMPLE 2.21. Let F =
GF(3)
and g = - y + xy+ x4
+y4 .
Then g satisfies conditions(1)-(3).
Note that(gx, gy)Fg[x, y]
nFg[x] = (X9 -
x +1)Fg[x].
It can be shown that theprime
factorizationof X9 - x
+ 1over
g
=GF(3)
isx9 - x
+ 1 =(x3 -
x +1)(x6 + x4 + x3 + X2
x -
1).
Thusby (2.1)
the class groupof Fg[x3, y3 , g]
is0, 7L/37L
or7L/37L
ffi7L/37L.
SinceD(gx)/gx
= 1 is in!£g, Cl(Ag)
is either7L/37L
or7L/37L
C7L/37L.
This calculation can be verified as follows.
(1.3)-(1.6)
are used to cal-culate
L,
thelogarithmic
derivatives of D inF[x, y].
ThenYg
= L nFg [x, y].
Thus t E L if andonly
if t = aoo + aiox + aoi y +03B120x2
+03B102y2
where
By eliminating
variables we find that03B10035 - 03B10034
+03B10032 + 03B1003-
03B100 = 0 and that the rest of the 03B1ijdepend
on aoo. Therefore the order of L is35.
Alsoif 03B1003 =
aoo then allother 03B1ij
= 0. ThusLg
is of order 3generated by t
= 1.2.23. For more details on how to
explicitly
calculate Y the reader is referredto [9], [10], [11] and [12] .
This next result refines the upper bound in
(2.19) slightly.
COROLLARY 2.24. Let g
satisfy
conditions(1) - (3) .
Let(f(x)) = (gx, gy) Fg[x, y]
nFg [x]. Let f(x) = f1(x) f2(x) ... fr(x) be a prime factorization of f(x) in Fg[x]
suchthat for
some s =1,
..., r,deg fi
+deg f2
+ ... +deg fs
>(n - 1) (n - 2)
where n =deg (g).
Then the orderof Cl(Ag) ps .
Proof.
Uses the sametype
ofargument
used in(2.19)
and the resultof (2.9).
EXAMPLE 2.25. Let F =
GF(3),
g = xy + X4 +y4.
Then gx = y +X3
gy = x +
y3 and H =
1. Then g satisfies(1)-(3). f(x)
=x9 -
x =(x2 -
x -
1)(x2
+ x -(x2
+1)(x
+1)(x - 1)x. By (2.24)
the order ofCl(Ag) 34.
This can be verified
by
directcomputation
ofLg.
One finds that L is oforder
35 generated by 1, x -
y, ax -a3 y, X2
+yz , ax2
+a3y2
wherea E
GF(9) - GF(3). Thus,
infact, Y9
and thereforeCI(Ag)
is of order33.
REMARK 2.26.
If g satisfies
conditions(l)-(4),
thenCl(Ag) ~ Z12Z if p
= 2and
Cl(Ag)
= 0 or7L/p7L
if p > 2.Example (2.15)
shows that these con-ditions are
not enough to
insure thatCl(Ag)
= 0if p
> 2. The next theorem adds one morecondition,
that appears to be not ageneric
one, that guaran- tees thatCl(Ag)
= 0.THEOREM 2.27.
Let g satisfy
conditions(1)
and(2). If for
each(03B1, f3)
ESg,
H(03B1,03B2) ~ Fg(a, 03B2).
ThenCI(Ag)
= 0.Proof:
IfCl(Ag) :0
0 thenby (2.11)
there exists(a, f3)
ESg,
t ELg
such thatt(a, b)
=n H(x, 03B2)
for some n ~ 0 inZ/pZ.
Since t 6Fg[x, y],
this is acontradiction.
COROLLARY 2.28.
Let g satisfy
conditions(1) - (4). Suppose
also that no twoelements
of Sg
have the samey-coordinate. If fôr
some(oc, f3)
ESg, .J H( et, 03B2) ~ Fg(03B1)
thenCl(Ag)
= 0.Proof.
Let(a, b)
ESg.
Then there is anFg-automorphism
ofP
that maps(oc, b)
to(oc, fi). If H(a, b)
EFg (a)
=Fg(a, b) by (2.5),
thena H(a, b)
EFg(03B1).
But
(03C3H(a, b))2
=aH(a, b)
=H(a, fi).
Thisimplies
that03C3H(a, b) =
± H(03B1,03B2) ~ Fg(03B1).
A contradiction.REMARK 2.29. There are two reasons
why
thehypothesis
of(2.27)
appears to be not ageneric
one. The first is that in calculations 1 found that this condition appears to hold about half the time. Thesecond,
and thismight explain
thefirst,
is that for any finitefield, GF(pm), (p"’
+1)/2
elements of it have a square root inGF( p"’ ).
EXAMPLE 2.30. Let p = 3 and g
= x2
+y2. Then gx
=2x, gy
=2y
and H = 2. The conditions of
(2.27)
areeasily
seen to hold. ThereforeCl(Ag)
= 0. This is verifiedby
the fact that Y is of order threegenerated
by 2 ~ Fg
=GF(3).
REMARK 2.31. The next two results were
proved by
Blass[3]. Although
in theintroduction to his article Blass assumes that the
degree
of g is divisibleby
p, the
proofs
of these results areindependent
of thisassumption.
LEMMA 2.32. Let g be as in
(2.16)
andp
5. Then the Galois groupof k(T¡j)/k(T¡j)
acts as thefull symmetric
group onSg (see [3)
page10).
LEMMA 2.33. Let g be as in
(2.16)
andp
5. LetQ1 ~ Q2
ESg.
Thenthere exists an
automorphism
u EGal((k(Tij)/k(Tij))
such that03C3(H(Q1)) = - (H(Q1), 03C3(H(Q2)) = -.JH(Q2)’ a H(Q)
=H(Q) for
allQ
ESg
withQ ~ QI, Q2
and such that u act as theidentity of Sg (see f3 J
page10).
EXAMPLE 2.34. Let g =
03A3Tijxiyj
be as in(2.16).
Then thering Ag
=k(Tij) [xp, yP, g]
is factorial wherep
5. This result followsimmediately
from
(2.27)
and(2.33).
3.
Properties
ofCI(AI)
REMARK 3.1. Before
moving
on to the main section of thisarticle,
somegeneral
facts aboutCl(Ag)
should be mentioned. First ofall,
we have thatif A =
F[xP, yP, g],
thenCl(Ag) injects
intoCI(A).
Thesimplest
way to seethis,
is to observe thatCl(A) - 2, Cl(Ag) - Lg
and thatLg
2. Thenany
general
statements that can be made aboutCI(A) concerning order, type,
etc., can also be made aboutCl(Ag).
In[11]
thefollowing
results wereproved
forCI(A)
which therefore alsoapply
forCl(Ag).
THEOREM 3.2. Let g
satisfy
conditions(1)
and(2).
ThenCI(Fg)
is a p-groupof type (p, ... , p) of order p"’,
where mdeg(g)(deg(g) - 1)/2 (see [11]
page
397).
THEOREM 3.3. Let g
satisfy
conditions(1)
and(2).
For eachpositive integer
n, let
A(n)
=Fg[xpn, yP", g]. Then,
(a) for
each n,Cl(A (n» injects
intoCl(Ag(n+1)),
(b) for
each n,Cl(Ag(n))
is a p-groupof
type(pil,
...,pir)
whereeach ij
n,(c)
the orderof Cl(A 9 (n»
=pf,
wheref n(deg(g))(deg(g - 1)/2 ([11]
page
406).
4. The main theorem
This section
begins by presenting
a newalgorithm (see [12]
page247)
forcomputing
the divisor class group of a Zariskiring
A =k[xP, yP, g]
definedover an
algebraically
closed field k of characteristicp ~
0.Then Theorem
(4.14)
proves that thering k(Tij)[xp, yP, g], where g =
03A3Tijxiyj is
as inexample (2.16),
is factorial. P. Blassproved
this result for thecase
deg(g) =
0(mod p)
in[3].
The
algorithm
and Theorem(4.14)
are then combined to prove that fora
generic
g, thering A
is factorial.4.1. Let k be an
algebraically
closed field of characteristicp ~
0. Letg E
k[x, y] satisfy
condition(1).
Thenby (1.8), Cl(k[xP, yP, g])
isisomorphic
to
2,
the additive group oflogarithmic
derivatives of D =gy(%x) -
gx(~/~y)
ink[x, y].
If t Ek[x, y]
is in L thenby (1.8), deg(t) n -
2 wheren =
deg(g). Furthermore, t
is in L if andonly if Dp-1 t
- ct = - tp whereDP = cD.
By (1.6)
it follows that t is in L if andonly
if(4.2) (a) V ( G’ t) =
0 for r =0, 1,
..., p -2,
and(b) ~(Gp-1 t)
-tP,
where V =alp-2laXp-layp-1.
Thus the elements of L can be determined in the
following
way.Let t =
03A30i+jn-203B1ijxiyj
be apolynomial
in x and y with undetermined coefficient. Substitute t into(4.2a)
and(4.2b)
and compare coefficients.When t is substituted into
(4.2a)
one obtains linearexpressions
in the aiiwith coefficients in
k,
sayls
=0,
0 s m with m anonnegative integer.
When t is substituted into
(4.2b)
one obtainsp-linear equations
of the formlij(03B1) = 03B1pij, 0
i+ j n - 2,
where1,y(a)
is a linearexpression
in the aij with coefficients in k.Thus it is
readily
seen that 2 isisomorphic
to the additive group of solutions to thep-linear system
ofequations
ls
=0, 0 s n
and1,j(a) = 03B1ijp, 0 i + j n -
2.(4.3)
In
[12]
analgorithm
forcomputing
the number of solutions to asystem
suchas
(4.3)
was described.What follows is a
description
of anotheralgorithm
which better suits the purposes of this article.Let N =
n(n - 1)/2.
let C be the coefficient matrix of the linear expres- sionslij, 0
i+ j % n -
2. Then C is an Nby
N square matrix.Assume first of all that det C ~ 0. Then each linear
expression ls
with0 s m
can beexpressed
as a linear combination ofthe lij
with coef-ficients in k. Thus
beginning with h
there exists a;, ,0 i + j %
N such that03A3aijlij
=l, .
Since11 (a) = 0,
this leads to03A3aij03B1ijp
=0,
which results in the linearequation h : 03A3aij(l/p)03B1ij
= 0. Thus for each s,0
s m, another linearequation ls’, 0 s
m, isproduced.
From these 2m linearequations,
choose a basis
l’’1, l’’2,
... ,1,*,"
where 0 u 2m. Nowrepeat
the firststep
ofgenerating
linearequations by writing
eachl’’s, 0 s
u, as a linearcombination of the
li,.
From these 2u linearequations,
choose a basis andcontinue this process. One of two
possibilities
will takeplace. One,
is thatat some
point
Nindependent
linearequations
will beproduced
in Nunknowns. If this is the case then each ocii = 0 which
implies
that 2 = 0.The alternative to this situation is that at some
point R linearly indepen-
dent
equations
will beproduced
and no more thanthat,
with R N.Any
new
equations produced
will be a linear combination of these Rindependent equations.
If this is the case then the number of solutions to thesystem (4.3) is pN-R .
To seethis,
choose N - Rp-linear expressions
from theequations lij
=ap
so that the linearpart
of theseequations together
with the R linearequations
form a k-basis for the space of all linearexpressions
in the aii with coefficients in k. This can be done since thelij
are a basis for this space. It then follows that thesystem
ofequations consisting
of these R linear and N - Rp-linear equations
isequivalent
to theoriginal system (4.3).
For ifild
-apd
is one of thep-linear equations
in(4.3)
thenlcd
is a linear combi- nation of the linearexpressions
in the N - Rp-linear equations
and the Rconstructed linear
equations.
It then follows that03B1cdp
is a linear combination ofthe ap
that appear in the N - Rp-linear equations.
This of course leadsto another linear
equation
aftertaking p-th
roots which mustby assumption
be
dependent
on the P linearexpressions.
It then follows from Bezout’s theorem that there arepN-R
solutions(see 4.4) below).
If it turns out that det C =
0,
where C is the coefficient matrix of the linearexpressions lij
in(4.3),
than the rank of C = N - M for some M > 0.Therefore from the
equations 1,, = 03B1ijp, 0
i+ j % n - 2,
one canimmediately generate
M linearequations.
These M linearequations
are thencombined with the m linear
equations ls
=0,
0 s m, and a basis forthe linear
equations
is chosen. At thispoint
there are N - Mp-linear equations
whose linear parts arelinearly independent
and somelinearly independent
linearequations.
If these linearexpressions (from
the N - Mp-linear equations
and the linearequations)
aredependent
then some non-trivial linear combinations of these
expressions
are 0. Asabove,
thesecombinations will
produce
nontrivialhomogeneous
linearequations.
Abasis for the linear
equations
is then chosen and combined with thep-linear equations
to form asystem
that isequivalent
to theoriginal system (4.3).
This process is
repeated
until one of twopossibilities
occurs. Either Nindependent
linearequations
will beproduced
in which case L = 0 or Rlinearly independent
linearequations
will beproduced
where R N andwhere the linear
expressions
from thep-linear equations
and the R linearequations
cannot be used toproduce
any new linearequations
that areindependent
from theexisting
linearhomogeneous equations.
If this is thecase then L is
of order pN-R .
To see this consider the k-vector spacespanned by
the linearexpressions
in thesep-linear equations
and in the Rlinearly independent homogeneous
linearequations.
Then a basis for this space canbe constructed that includes the R
linearly independent
linearequations.
Then
arguing
as above one sees that thesystem
ofequations consisting
ofthe R linear
equations
and thosep-linear equations
used to construct the basis isequivalent
to thesystem (4.3).
Thisequivalent system
must consist of a total of Nequations
otherwise there would be more unknowns thanequations
and hence an infinite number ofsolutions,
which wouldimply
thatj2f is infinite. This contradicts
(3.1).
Therefore anequivalent system
ofN -
R p-linear
and R linearequations
in N unknowns has been constructed with theseproperties
that are easy toverify:
(4.4) (a)
There are no intersections atinfinity,
and(b)
Themultiplicity
of eachpoint
of intersection is one.Then
by
Bezout’s theorem the total number of intersectionpoints
ispN-R.
This then is the
algorithm
fordetermining
the order of 2.REMARK 4.5.
Although
thisalgorithm
is much moreclumsy
than thealgorithm
in[12]
forcomputing
the divisor class group of A =k[xP, yP, g],
it proves very useful in
determining Cl(A)
for ageneric
g.EXAMPLE 4.6. Let k be an
algebraically
closed field of characteristic 3 andg = x + y + x5 + y5 . Applying
thisalgorithm
one finds thatCl(A)
isisomorphic
to the additive group of solutions to thesystem
This system is
easily
seen to beequivalent
to thesystem
In the first
step
of thealgorithm (with
det C ~0)
one obtains the linearequations
In the next
step
no newindependent equations
areproduced.
Thus the orderof Cl(A) is p8-2 = 36.
The most
important application
of thisalgorithm
is the next result.THEOREM 4.10. Let k be an
algebraically closed field of
characteristicp ~ 0,
n 4 be a
positive integer, {Tij: 0
i+ j nl
be a setof
indeterminatesover
k,
F =k(Tij)
and g =03A30i+jn Tijxiyj. If Cl(k(Tij)[xp, yP, h]) ~
0then
Cl(k[xP, yP, g]) ~
0for
ageneric
choiceof coefficients
aij E kof g =
03A30i+jnaijxiyj.
Proof.
Assume thatCl(k(Ti ) [xP, yP, g])
is 0. When thealgorithm
in(4.1)
isapplied
to g, we arrive at thesystem
ofequations (4.3) consisting of p-linear
and linear
equations
with coefficients in thepolynomial ring GF( p) [T¡j].
Inthe next
step
of thealgorithm
additional linearequations
aregenerated,
thistime with coefficients in
[GF( p) (Tij)](l/p)
where for a field L of characteristicp ~ 0, L(l/pn)
is the field of all elements a E I such that apn E L. In the m-thstep,
more linearhomogeneous equations
aregenerated
with coefficients in the field[GF( p) (Tij)](l/pm).
The class group ofk(Tij) [xP, yP, g]
is trivial if andonly
ifeventually
Nlinearly independent homogeneous
linearequations
inN unknowns are
generated by
thisalgorithm,
N =(n - 1)n/2.
Thatis,
ifand
only
if Nhomogeneous
linearequations
in N unknowns aregenerated
with coefficient matrix B such that