KAWAI, OSAMU

MIYANISHI, MASAYOSHI

Osaka City University Repository

Kawai, O. and Miyanishi, M.
Osaka J. Math.
23 (1986), 207-215
SPECIALIZATIONS OF COFINITE SUBALGEBRAS
OF A POLYNOMIAL RING
Dedicated to Professor Hirosi Nagao on his sixtieth birthday
OSAMU KAWAI AND MASAYOSHI MIYANISHI
(Received December 14, 1984)
1. Introduction. Let K be a field of characteristic zero and let R
κ
:=K[x, y] be a polynomial ring in two variables over K. A normal ^-sub-
algebra A of R
κ
 is said to be cofinite if R
κ
 is a finite ^4-module with the canonical
^4-module structure. In the case where K is an algebraically closed field,
we know the following results:
(1) If A is regular, A is then a polynomial ring in two variables over K;
see [3] and [8].
(2) If A is singular, then there exist a polynomial subalgebra R'
κ
 and
a finite group G of linear X-automorphisms of R'
κ
 such that A=(R'K)G and
G is a small subgroup of GL(2, K); see [4] and [10].
In the present article, we shall show that the structures of normal cofinite
subalgebras A of R
κ
 are invariant under specializations, provided the quotient
field extension Q(RK)/Q(A) is a quasi-Galois extension; see Definition 2.2.
Our problem is formulated as follows: Let ©=&[[£]] be a formal power series
ring in one variable over an algebraically closed field of characteristic zero and
let R:=Ό[x, y] be a polynomial ring in two variables over ©. Let A be an
©-subalgebra of R. We say that A is cofinite if R is a finite ^4-module and
that A is geometrically O-normal if AK\=A®K and Ah\=A\tA are normal
D
domains, where K is the quotient field Q(©) of O. If A is a cofinite, geometric-
ally ©-normal subalgebra of R, then A
κ
 and Ak are cofinite normal subalgebras
in R
κ
 and Rk, respectively. Let K be an algebraic closure of K, We ask wheth-
er or not certain properties of a cofinite normal subalgebra Ajt of Rjt are in-
herited by the cofinite normal subalgebra Ak of Rk. We pose the following
Conjecture 1. Let © and Rbe as above, and let Abe a cofinite, geometrical-
ly ©-normal subalgebra of R. Then there exist a cofinite Ό-subalgebra Rr of R and
a finite group G of Ό-automorphisms of Rf such that:
(i) R' is a polynomial ring in two variables over © and contains A as an
O-subalgebra;
208 O. KAWAI AND M. MIYANISHI
(ii) A is the G-invariant subalgebra {R')G of R'.
Our result, though partial, is the following:
Main Theorem. Let O, R, K and K be as above. Let A be a normal,
cofinite O-subalgebra of R. Suppose that Q(R) is a quasi-Galois extension of
Q(A) over K. Let G be the Galois group of the extension Q(R)®KlQ(A)®K.
Then the following assertions hold true:
(1) G acts effectively on R, and A=RG. Namely, R is a Galois extension
of A with group G in the sense of [11],
(2) A is geometrically O-normal.
(3) Rk is a Galois extenύon of Ak with group G.
(4) If AR is a polynomial ring in two variables over K, so is Ak over k.
We shall see later that Conjecture 1 is reduced to the following:
Conjecture 2. Let O and R be as above. Let A be a normal, cofinite
O-subalgebra of R such that A
κ
 is a polynomial ring over K. Then Ak is a poly-
nomial ring over k; hence A is a polynomial ring over O by virtue of a result of
Sathaye [14]; see also Kambayashi [6],
Concerning the second conjecture, we can show that Spec Ak has at most
one singular point which has necessarily cyclic quotient singularity, provided
A is geometrically O-normal; see Proposition 4.1 below.
2. Representability of a group functor
Let K be a field of characteristic zero, let L be a regular extension of K
and let L' be a finite algebraic extension of L. Suppose that L' is a regular
extension of K.
Let C be the category of finite, reduced i£-algebras. We define a group
functor Aut
κ
(L'IL) on the dual category C° by
Spec(S)<=£° h-» Aut
κ
(L'IL) (S):= Aut(L'®SIL®S),
K K
where KvX(L'®SjL®S) denotes the group of all L®S-algebra automorphisms
K K K
of L'®S, which is a finite group. We then have the following:
K.
Lemma 2.1. The functor Aut
κ
(Lf/L) is representable by a finite group
scheme over K.
Proof. Let X be a projective normal variety defined over K such that
L=K(X) and let Xf be the normalization of X in V. Let v\ X'->X be the
normalization morphism. We define a group functor A\it
κ
(X'jX) on the
category of ίΓ-schemes by
SPECIALIZATIONS OF COFINITE SUBALGEBRAS 209
/iQ H» Aut
κ
(X'IX) (T):= Aut(X'x T/Xx T),
K K
where Aut(X'xT/XxT) denotes the group of all Xx Γ-automorphisms of
K K K
X' X T. We claim that the restriction of Aut
κ
(X'IX) on the full subcategory
K
C° of (SchjK) coincides with the group functor A\xtK(L'/L)0 which is the op-
posite of AutK(X'/X)9 i e > ^ e order of multiplication is reversed.
In fact, let S be a finite, reduced i£-algebra. Then S is a direct product
n
S= Π Ki, where K( is a finite algebraic extension of K. We have apparently
i = l
Aut
κ
(X'IX)(S) = Π Aut(X'®KiIX®Ki), and
ι = l K K
Ant
κ
(L'IL) (S) = Π Aut(L' ® KJL ®K{).
i—1 K K
Hence we may (and shall) assume that S is a field. Note that X®S is a normal
K
variety and X'®S is the normalization of X®S in the field L'®S. More-
K K K
over, it is easy to show that the canonical homomorphism
Aπt{X'®SjX®Sγ -> Ant{L'®SjL®S)
K K K K
is an isomorphism.
Now, applying the representability criterion of Grothendieck [2; 221-19],
Aut^X'/^Q i s representable by a i£-group scheme, say Aut
κ
(X'/X), which is
locally of finite type over Kι\ However, since \AutK(X'/X)(Kf)\^[Lf: L]
for any finite algebraic extension K' of K, Aut
κ
(X'jX) is a finite K-
group scheme. Moreover, since char(i£)=0, Aut
κ
(X'/X) is reduced by a
theorem of Cartier (cf. [12]). Therefore we know that Aut
κ
{L'jL) is repre-
sentable by a finite i^-group scheme Aut
κ
{X'\X)\ Q.E.D.
We denote AutK{X'jX)° by Autκ{LfjL) or simply by β. Write Q= Spec(^).
Then the identity morphism idg: Q-+Q corresponds to an L-homomorphism
1) Define a functor H o m ^ f Γ , X) on (Sch/K) by
ΓeCSch/iQ^HoinKtAr, X) ( T ) = H o m
ί P ( ^ , Xτ)
(cf. [2], [16]). Then there exists the canonical morphism of functors
Φ: AuUCXO-^HomjEiX', X)
such that, for T(=(SchlK) and tf(=Autr(XίO, 0r(a) = i/j"«. Note that both AutK(X') and
Hom.E;(X'> X) are representable by i^-schemes locally of finite type, and hence Φ is repre-
sentable by a morphism of i£-scheroes,
/: Aut
κ
{X') -» Hom
κ
{X', X)
(cf. [15] and [16; Th. 3]). The iC-scheme Hom
κ
{X',X) has a iC-rational point v. X'-+
X. It is now apparent that AutK{X'jX) is representable b y / " 1 ^ ) , which is a 2£-group
scheme locally of finite type.
210 O. KAWAI AND M. MIYANISHI
Δ: U -* L'®Jί,
K
and for any S&C and any element αGHom^_
a
ig(cJί, S)=Autκ{L'jL){S), the
action of a on L'®S is given by (idL'®a)A: L'-*L'®S. It is then easy to
K K
see that the homomorphism Δ defines an action of Ω on Spec L'
σ: Ωx Spec L' -> Spec L'
which is a Spec L-morphism. We denote by (L')& the set
which is a subfield of L' containing L.
DEFINITION 2.2. We say that L'jL is a quasi-Galois extension over K
if (L')£=L.
Let K' be a finite algebraic field extension of K. Then it is straight-
forward to show that:
(1) Aut
κ
,{L'®K'jL®K')c^Aut
κ
(L'jL)®K'.
K K K
(2) The action of Aut
κ
^Lr®KrjL®K') on Spec (L'®Kf) is given by
K K K
A®K': L'®K' -* (®)®(
K K I I' I
and we have {L'®K')β'={L')3®K', where Q'=Q®K'.
K K K
Lemma 2.3. The following conditions are equivalent:
(1) L'jL is a quasi-Galois extension over K.
(2) For any finite algebraic field extension K' of K, L'®K'IL®K' is a
quasi-Galois extension over Kf.
(3) L'®KjL®K is a Galois extension, where K is an algebraic closure of K.
K K
Proof. The equivalence of (1) and (2) is clear in view of the preceding
observations. (2) =^(3): There exists a finite algebraic extension K'jK such
that Ω'\= Q®K' is a constant iΓ-group scheme with group G:= Q(K'). Since
G=Aut(L'®KΊL®K') and (L'®K')G=L®K'y L'®K'IL®Kf is a Galois
K K K K K K
extension with group G. Hence L'®K"IL®K" is a Galois extension with
K K
group G for any field extension K" of K with K"^K'. (3)-*(l): The condi-
tion (3) implies that L'®K'IL^K' is a Galois extension for some finite alge-
K K
braic extension K'jK, Since L®K'={L')£®K' as noted above, we have
K K
(L')β-L. Namely, L'jL is a quasi-Galois extension over K. Q.E.D.
Corollary 2.4. L'jL is a quasi-Galois extension over K if and only if \Q\
SPECIALIZATIONS OF COFINΠE SUBALGEBRAS 211
(: = the rank of ^-module <Jl) is equal to [LΊ L],
A quasi-Galois extension is not necessarily a Galois extension as shown
by the following trivial
EXAMPLE. Let K be the rational number field Q, let L=K(x) with in-
determinate x and let L'=K(y), where y"=x and n>2. Then Q=Aut
κ
(L'\L)
^Spec Q[ξ]l(ξn—1) and S(Q)ΦZInZ. Hence L'\L is a quasi-Galois extension,
but not a Galois extension. In fact, let K' be the extension of Q with all w-th
roots of unity adjoined. Then Q(K')^ZjnZ and Lf®Kf/L®Kf is a Galois
extension.
We don't know which conditions on K assure that a quasi-Galois extension
L'jL over K is a Galois extension. In the next section, we shall, however,
show that this is the case if K is the quotient field of a formal power series ring
k[[t]] in one variable over an algebraically closed field k of characteristic zero.
We use only the property that k[[t]] is strictly henselian.
3. Constancy of the iΓ-group scheme PiMt
κ
{L'jL)
Let (O, tΌ) be a discrete valuation ring of equicharacteristic zero, let K
=Q(O) be the quotient field and let k be the residue field. First of all, we
shall prove:
Lemma 3.1. Let A be a finitely generated, normal O-domain and let L
=Q(A). Let U be a finite Galois extension of L with group G and let A' be the
integral closure of A in L'. Then the following assertions hold true:
(1) G acts effectively on A'k9 and the canonical injection Ak^>Afk induces an
isomorphism Ak—(A'k)G.
(2) Suppose A'k is an integral domain. Then Q(A'k) is a Galois extension of
Q(Ak) with group G.
Proof. Our proof consists of several steps.
(I) Note that A' is a finite A -module (cf. Matsumura [7]). Furthermore,
Ak is a subring of A'k. In fact, we have only to show that A(~)tA'=tA. Sup-
pose a=ta' with α E i and a'^A'. Then α 'GQ(i) and a' is integral over
A. Hence a'^A because A is normal. The Galois group G acts effectively
on Af and A=(A')G. Hence G acts on A'h and Ak^(A'k)G.
(II) We shall show that G acts effectively on A'k. Suppose, on the contrary,
that an element g^G of order n>ί acts trivially on Afk. For any element a'
we have
aί with
Write ga{=aί+ta/2 with a'2^A''. Inductively, we define a\^A' (ί^i^n) by
g
a'i-
m
ι—a\-ι-{-ta'i. Then it is easy to show
212 O. KAWAI AND M. MIYANISHI
a' = *V = a'+ntaί+ -
Hence # ί e k 4 ' . Namely, we can write ga'—a'-\-fa{'. This is true for every
By the same argument as above with t replaced by /2, we have a[f
Thus, we can show that V — α ' e ΓitmA'. Since A' is a Noetherian
integral domain, we have ΓltmA'=(0) by KrulΓs intersection theorem (cf. [11]).
Namely, ga'=a' for every a'&A'. This is a contradiction.
(Ill) We shall show that Ak=(A'k)G. In fact, suppose a'^(A!k)G, and write
ea' = a
r+tb(g) with b{g)<=A\
where α'G^ί' with tf'=α'(mod £4'). Then we have
b(hg) = hb(g)+b(h) for £ , Λ G Ξ G .
Set *=(Σδfe))/|(?|. Then b(g)=c-gc for any ^ G G , and a'+tcEί(Af)G=A.
gξC}Hence ar^Ak. Namely, we have Ak=(A'k)G. Now, the assertion (2) is readily
ascertained. Q.E.D.
Hereafter, we assume that O is a formal power series ring k[[t\] over an
algebraically closed field k of characteristic zero. The constancy of the K-
group scheme AutK(Lf/L) is assured by
Lemma 3.2. Let O=k[[t]] be as above and let K=Q(O). Let L be a
regular extension of K and let U be a quasi-Galois extension of L such that L'
is a regular extension of K. Then L'\L is a Galois extension.
Proof. We have only to prove that the jfiΓ-group scheme Aut
κ
(L'\L) is
constant. Since the Puiseux field Όk((t1/n)) is an algebraic closure of k((t)),
«>0
where k((t1/n)) is the quotient field of &[[*1/M]], there exists a cyclic extension
°
/ =
* [ H ] of O (τn=t) such that AutK(LfIL)®K'~AutK,{L'®KfIL®K') is
K K K
constant, where K'=Q(Ό'). Note that the morphism Spec O'->Sρec O is a
faithfully flat and finite morphism. Let G=Aut
κ
(Lf/L)(K'). Then the con-
stant K'-group scheme G
κ
> has apparently a Nόron model Go', a constant O'-
group scheme with group G. Hence the i£-grouρ scheme Aut
κ
(L'/L) has
an O-Nάron model Q\ see [13] for relevant results. By definition, the group
scheme Q is smooth over O and satisfies 3®K—Aut
κ
{LfjL). By virtue of
D
[1; IV (18.10.16)], S is finite and Stale over O. Therefore Q must be a con-
stant O-group scheme ifc, where H—Q(k)=G(K). Since G=Q{Kr)^H®{K')
=Hy we know that <2~GD. Thus L'\L is a Galois extension with group G.
Q.E.D.
SPECIALIZATIONS OF COFINITE SUBALGEBRAS 213
Lemma 3.3. Let the notations and the assumptions be the same as in Lemma
3.1. Assume that L is the quotient field of a finitely generated, normal O-domain
A. Let A' be the normalization of A in L\ and let G be the Galois group of the
extension L'jL. Then the following assertions hold true:
(1) A1 is a Galois extension of A with group G.
(2) Suppose that Ar is geometrically O-normal Then, so is A, and A'k
is a Galois extension of Ak with group G.
Proof. (1) is now clear. As for (2), Al is a normal domain by the hypo-
thesis, and Ak=(A'k)G by Lemma 3.1. Hence Ak is normal, and A is geo-
metrically O-normal. The remaining assertion is clear by Lemma 3.1. Q.E.D.
Now, Main Theorem except the assertion (4) follows from Lemma 3.3.
In fact, set L: = Q(A) and L': = Q(R) with A and R as in Main Theorem. Then
R is the normalization of A in Z/, and R is geometrically ©-normal. So, we
can apply Lemma 3.3. We shall prove the assertion (4). Since Ajt is a poly-
nomial ring over R> A
κ
 is a polynomial ring over K by [5]. We can identify
G as a finite subgroup of GL(2, K)y and it is well-known that G is then gener-
ated by pseudo-reflections. Recall that an element g^GL(2t K) is a pseudo-
reflection if and only if the fixed-point locus Y{g)ϊt in ^ : = Spec K[x,y] under
the action of g has codimension^l. Since g acts on .4.0:= Spec fθ[x,y], let
T(g) be the fixed-point locus in AQ under the action of g. Namely, T(g) is
a closed subscheme of A2D defined by an ideal /, where / is the smallest ideal
of O[#,jy] generated by all elements of the form ga—a with a^O[x,y]. Then
we know that T(g)κ=Γ(g)(£)R and that T(g)(g)k is the fixed-point locus in A2k: =
D D
Spec k[xf y] under the action of g. Hence Γ(g)(g)k has codimension^l in
Ah This implies that when one embeds G into GL(2> k) upto conjugation
in Antkk[x, y]> G is generated by pseudo-reflections. Hence the G-invariant
subring Ak of k[xy y] is a polynomial ring over k. This verifies the assertion
(4) of Main Theorem.
4. Reduction from Conjecture 1 to Conjecture 2
Let O, R and A be as in Conjecture 1. Let Y:=A20=Sρtc R, let X: =
Spec A and let π: Y-+X be the canonical finite morphism. For an algebraic
closure K of K=Q(Ό), Ajt is a normal, cofinite ^-subalgebra of ^[^,3;]. Note
that Xκ=Spec Aft has at most one singular point. Let Z' be the universal
covering space of Xκ~Sing(Ji^). Then UK Y&—π~ι (Sing Xκ)->Xχ—Sing(X%)
factors through Zf because YR—π~x (Sing XR) is simply connected. Let Z
be the normalization of XR in the function field R{Zf) of Z'. Then Z^A\
and πκ> YR-^XK factors through Z;
V a 7 β7TR' * K * £
214 O. KAWAI AND M. MIYANISHI
See [10] for the relevant results. Choose a ^-rational point P of YR—
TΓ-^Sing XK), and let Q=a(P). We shall show that Z descends down to a K-
scheme. Namely, there exist a X-scheme Z and X-morphisms a: Y—>Z and
β: Z-*X such that Z=Z®K, a=a®R and β=β®R. In fact, for σG
JS. K K
Gal(K/K), let σZ=Spec p
σ
(Θ(Z)), where p
σ
: R{x,y]-+K[x,y] is σ®id
κίXfyl and
O(Z) is the coordinate ring of Z which is a ^-subalgebra of R[x, y]. We denote
by σa: Yκ~->σZ and σj3: σZ-+Xκ the morphisms induced by ρ
σ
{O{Z))^R[x,y]
and Aκ^ρ
σ
(O(Z)), respectively. Hence
 πR=(σj3)-(σa). Let %) be the point
of σZ which corresponds to Q under the canonical isomorphism Spec p
σ
(O(Z))
—> Spec(3(Z). Then we have a unique ./^-isomorphism φ
σ
:
 σZ-+Z such that
pj^Q)z=Q9 a=φσ.σa and σβ=β φσm Then it is easy to show that φ τ σ =
φ
τ
 τ
φ
σ
 for σ, r^Gdλ{RjK). In fact, this is the case for a finite Galois extension
K'jK instead of K/K. By the faithfully flat descent, we know that there exists
aX-schemeZsuchthatZ=Z®R. T h e n c e σ a a n d ^ ^ y S f
Therefore a and B descend down to i£-morρhisms a: XK"^ Z and β: Z-> Yκ
such that π
κ
—β a. On the other hand, Z is X-isomorphic to A\ by virtue of
[5]. Identify the coordinate ring O(Z) with a i£-subalgebra of K[x, y] under
a. Let B be the normalization of A in the function field K(Z) of Z. Then
B is a normal, cofinite O-subalgebra of R such that BK=O(Z) is a polynomial
ring over K. The Conjecture 2 then implies that B is a polynomial ring in
two variables over O. Note that Q(B) is a quasi-Galois extension of Q(A)
over K. Main Theorem then asserts that Conjecture 1 is affirmative.
As for the Conjecture 2, we know the following:
Proposition 4.1. Let D, i? ##*/ A be the same as in Conjecture 2, and let
X ^ S p e c A. Suppose that A is geometrically ©-normal. Then Xk has at most
one singular point which has necessarily cyclic quotient singularity.
Proof. By the hypothesis, A
κ
 is a polynomial ring K\u, v\. Let Δ = — ,
du
which is a locally nilpotent ^-derivation of A
κ
. Since A is finitely generated
over ©, we find an integer n^O such that tnA(A)^A and fA(A)QtA. Define
a ^-derivation δ of Ak by
8{μ) = fA(a) (mod tA),
where a—α(mod ί^ 4) with a^A. Then 8 is well-defined, and δ is a nontrivial,
locally nilpotent ^-derivation on Ak. Hence Xk:= Spec Ak is afKne-ruled
(cf. [9]) and Xk has at most one singular point which has necessarily cyclic
quotient singularity (cf. [10]). Q.E.D.
SPECIALIZATIONS OF COFINITE SUBALGBBRAS 215
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Department of Mathematics
Osaka University
Toyonaka, Osaka 560
Japan



Specializations of cofinite subalgebras of a polynomial ring

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