Graded factorial domains - J-STAGE

Japan. J. Math.

Vol. 3, No. 2, 1977

Graded factorial domains

By Shigefumi M0RI

(Received February 14, 1976)

Introduction

We are concerned with a graded domain R such that R is finitely gen

erated over a field k, R0 is a field (hence it is a finite extension held of k),

and Rn=0 if n<0. By modifying gradation, we may assume that R satisfies

the condition G.C.M. {n•¸N|Rn•‚0}=1. In this case, the graded k-algebra

R is said to be almost-geometric (Definition 2.1). The purpose of this paper

is to classify almost-geometric graded factorial domains R over k by their

geometric data. If dim R=1, it is clear that R_??_R0[X], where deg X=1.

If dim R=2 and k is algebraically closed, then our classification is stated

as follows (Theorem 5.1 and Remark 5.2):

Let r be the minimum of numbers of homogeneous elements which gen

erate R. Then:

a) If r=2, then R_??_k[X, Y], where (deg X, deg Y)=1.

b) If r_??_3, then the set of isomorphism classes of such graded k

algebras R is in one to one correspondence with the set of pairs (a, e) such

that e is an r-ple of natural numbers e1, •c , er (e1>•c>er>1, and (ei, ej)

=1 if i•‚j) and a is an (r-2)-plc of mutually distinct elements a3=1, a4,•c

, ar of k*. The correspondence is defined by

R_??_ k[X1,•c , Xr]/(Xe11+a3Xe22+Xe33,•c , Xe11+arXe22+Xerr),

where deg Xi=(„Prj=1 ej)/ei (i=1, •c r).

In order to obtain this result and corresponding results in higher

dimensional cases, we have to generalize the results of our previous paper

[6, •˜1]. We construct an almost-geometric graded domain R(X, L) defined

by a semi-complete polarized k-variety (X, L), and we call such a graded k

algebra geometric (•˜1 and •˜2). For an almost-geometric graded factorial

domain R over k of dimension_??_2, there is a unique natural number m (called

the index of R) such that R(m) (for the notation, see the end of "Notation

and terminolygy" below) is a geometric graded factorial domain (Theorem

224 SHIGEFUMI MORI

3.1 and Corollary 3.3). R is obtained as R(m) [v1/e] from R(m) and a ramifi

cation data (v, e) over R(m) (Theorem 4.4). Thus these graded k-algebras R

are classified by semicomplete polarized k-varieties (X, L) and ramification

data over R(X, L) such that X is locally factorial, dim X>0, and Pic X=

Zcl(L) (Corollary 1.10 and Theorem 4.5).The author expresses his hearty thanks to Professor K. Watanabe of

Tokyo Metropolitan University for valuable discussions. Indeed, the start

ing point of our study was the special case r ® 3 of the assertion (b) men

tioned above, which Professor K. Watanabe first proved using the theory

of semigroup rings and announced at the meeting of the Mathematical

Society of Japan in the fall of 1975. The author also expresses his hearty

thanks to Professor . Hartshorne for his useful advices and giving us a

cohomological proof (different from ours) of Theorem 1.2 which is based on

the coherence of a sheaf of local cohomology groups (cf. [5, Expose VIII]).

Nation and terminology

Throughout this paper, we fix a field k and follow generally the notation

of [2]. By a ring we understand a commutative ring with identity element,

and by a module a unitary one. We understand by a k-variety an irreducible

reduced separated algebraic k-scheme, and by a polarized k-variety (X, L)

a pair of a k-variety and an ample invertible sheaf L on X. By the line

bundle V(L) associated to an invertible sheaf L on X, we understand the one

introduced in [2].

Symbols *, Q, Cl, and Pic are used as follows: A* denotes the group

of the units of a ring A, Q(A) (or Q(X)) denotes the quotient field of an in

tegral domain A (or, an integral scheme X, resp.), Cl A (or, Cl X) denotes

the divisor class group of a normal noetherian domain A (or, a normal no

etherian scheme X, resp.), and Pic X denotes the abstract group of isomor

phism classes of invertible sheaves on a noetherian scheme X. For a scheme

X and a quasi-coherent sheaf F on X, we denote by R(X, F) the graded ring

_??_i_??_0 H0(X, Si(F)) such that every element of H0(X, Si(F)) is of degree i.

For a graded ring R, we denote by Ri the homogeneous part of degree

i of R for every i•¸Z and, in case Ri=0 for all i<0, we denote by R+ the

homogeneous ideal _??_i>0 Ri; in case R is an integral domain, we denote by

QH(R) the quotient ring M-1R, where M is the multiplicative set of non-zero

homogeneous elements of R. For a natural number n, R[n] denotes the

graded subring _??_i•¸z Rni of R, i.e. R[n]i=Ri (or, 0) if i•ß0 mod n (or,

i_??_0 mod n, resp.). For a non-zero rational number q, R(q) denotes the

graded ring _??_i•¸Z R(q)i, Where R(q)i=Rqi (or, 0) if qi•¸Z (or, qi•¸Z, resp.), with

the natural ring structure. (If q•¸N, this graded ring R (q) coincides with the

Graded factorial domains 225

one defined by [2].)

•˜ 1. Graded rims associated with semi-complete polarized k-varieties

In this section, we study the graded ring R(X, L) associated with a

polarized k-variety (X, L) assuming the "semi-completeness" of (X, L) in

stead of the propernees of X. The results of this section are simple generali

zations of those of our paper [6, •˜1].

DEFINITION 1.1. A porarized k-variety (X, L) is said to be semi

complete if there are a natural number n and an embedding ƒÓ of X in some

projective space PNk such that ƒÓ*_??_pNk (1)_??_L_??_n and codimx (X-X)_??_2, where

X is the scheme-theoretic closure of X in PNk.

It is clear that, for an arbitrary natural number m, (X, L) is semi

complete if and only if (X, L_??_m) is semi-complete (cf. Example 1.12).

From now on until the end of this section, we fix a semi-complete

polarized k-variety (X, L).

THEOREM 1.2. R(X, L) is a finitely generated graded domain over k,

dim R(X, L)=dim X+1 and R(X, L)0 is a finite extension field of k (cf. [6,

Proposition 1.1]).

PROOF. We may assume that k is an infinite field. Indeed, let k(t) be

a purely transcendental extension of k, then since (X, L) is semi-complete,

(Xk(t), Lk(t)) is semi-complete by the definition, and obviously R(Xk(t), Lk(t))=

R(X, L)_??_k k(t). Hence if R(Xk(t), Lk(t)) enjoys the properties stated in the

theorem, so does R(X, L) (theory of descent, cf. [4, Expose VIII, Corollaire

3.4]). Thus we have reduced to the case where k is an infinite field. Let n

and ƒÓ:X•¨PNk be as in Definition 1.1. Since k is an infinite field, we obtain

a finite dominating morphism ƒµ:X•¨Prk (r=dim X) by means of a suitable

projection PNk•¨Prk. Hence ƒµ _??_Pk(1)_??_L_??_n. Setting U=Prk-ƒµ(X-X), we

obtain the following commutative diagram of natural homomorphisms.

Obviously ƒ¿, ƒÀ, ƒÁ, ƒÂand are injections. ƒÀ is an isomorphism because

depth (Prk-U) _??_Prk=codim Pri (Prk-U)_??_2 (cf. [3, Theorem 3.8]). By Lemma 1.3

below, ƒÁ is an integral extension, hence so is ƒÃa. Since Q(R(X, L)) is a

fi nite extension of Q(R(Prk, _??_(1))) (of degree n[Q(X): Q(Prk)]) and since R(Prk,

_??_(1)) is a polynomial ring in (r+1) variables over k, the integral closure of

226 SHIGEFUMI MORI

R(Prk, _??_(1)) in Q(R(X, L)) is a finite R(Prk, _??_(1))-module [1, Chapitre V, •˜1, n•‹2,

Theoreme 2]. Hence ƒÃƒ¿ is a finite extension. Thus the theorem is proved.

LEMMA 1.3. Let f: Z•¨Y be a finite dominating morphism of k-varie

ties, and M an invertible sheaf on Y. If Y is normal, then R(Z, N) is an

integral extension of R(Y, M), where N=f*M.

PROOF. Since R(Z, N)=ƒ¡(V(N), _??_v(N,) and R(Y, M)=ƒ¡(V(M), _??_v(M)),

we have only to prove that P(Z, _??_Z) is integral over P(Y, _??_Y). Let x be an

arbitrary element of P(Z, _??_Z), and let Tn+c1Tn-1+ •c+cn be the minimal

polynomial for x over Q(Y) (ci•¸Q(Y) (i=1, •c, n)). It suffices to prove that

ci•¸ƒ¡(Y, _??_Y) for every i, and we may assume that Y is affine, namely Z=

Spec R and Y=Spec A, where R is a finite A-algebra. Then ci•¸A (i=1,

•c, n) by the normality of A [1, Chapitre V, •˜1, n•‹3, the corollary to

Proposition 11]. q.e.d.

By the same proof as above, we obtain

COROLLARY 1.4. R (X , L•É)=H0 (X, _??_x) if dim X>0.

Now, for convenience, we quote the following result (in a simplified

form) from [2, Proposition (8.8.2)].

PROPOSITION 1.5. Let R=R(X, L). Then there are an open immersion ƒµ

: X•¨Proj R and a morphism g: V(L)•¨Spec R such that (i) g contracts the

zero-section (denoted simply by X) of V(L) to the point {R+}, (ii) g induces

an open immersion g': V(L) -X•¨Spec R-{R+}, and (iii) the following

diagram commutes.

Furthermore *CQ(n) L_??_n for even integer n.

By virtue of the above proposition, we obtain most of the following

results in the same way as in our paper [6, •˜1], and we sketch or omit proofs

of them.

COROLLARY 1.6. R(X, L) is normal if and only if X is normal (cf. [6,

Proposition 1.2]).

COROLLARY 1.7. If dim X>0, then Spec R-g'(V(L)-X) is of depth_??_2

in Spec R, i.e. even point of Spec R-g'(V(L)-X) is of depth_??_2 in Spec R

(cf. [6, Proposition 1.7]).


PROOF. By the definition of depth, this corollary is equivalent to the

assertion that the restriction map R=ƒ¡(Spec R, _??_spec R)•¨F(V(L)-X, _??_V(L)-x)

=_??_n•¸Z H0(X, L_??_n) is bijective, and this follows from Corollary 1.4. q.e.d.

COROLLARY 1.8. Proj R-ƒµ(x) is of depth_??_2 in Proj R.

PROOF. We may assume that R is generated by R1 as an R0-algebra, by

replacing L with some multiple of L (Theorem 1.2). Then the vertical

morphisms in the diagram of Proposition 1.5 are Gm-bundles. Hence our

assertion is proved by Corollary 1.7. q.e.d.

THEOREM 1.9. Let us assume that X is normal and dim X>0. Then

we have the following exact sequence:

0•¨Z•¨Cl X•¨C1 R(X, L)•¨0,

where 1(•¸Z) is mapped to cl(L), the class of L in Cl X (cf. [6, Theorem 1.3]).

PROOF. The proof of [6, Theorem 1.3] can be applied easily to our case

by virtue of Corollary 1.7 and of the isomorphism Cl X•¨C1 V(L) (cf. [6,

(1.5)] and [5, Expose X, Corollaire 3.8]). q.e.d.

With the notation as above, if Cl X is generated by cl(L), then X is

locally factorial. Thus we have

COROLLARY 1.10. R(X, L) is factorial if and only if X is locally factorial

and Pic X is isomorphic to Z and is generated by cl(L).

Since some positive multiple of L extends to an ample invertible sheaf

on Proj R(X, L), the immersion ƒµ: X•¨Proj R(X, L) is universal in the

following sense:

REMARK 1.11. Let X_??_X be as in Definition 1.1. Then this mor

phism is expressed uniquely as a composite morphism

X•¨ƒµProj R(X, L)•¨fX

where f is a finite dominating morphism.Indeed, the proof of Theorem 1.2 shows that R(X, L) is finite over R(X,

_??_X(1)). Thus we obtain the required morphism f of Proj R(X, L) to X.Before closing this section, we give an example which shows that the

semi-completeness of a polarized variety (V, F) depends not only on V but also on F.

EXAMPLE 1.12. Let us assume that k is algebraically closed and of in

228 SHIGEFUMI MORI

f inite transcendence degree over its prime subfield. Let Y be a complete

non-singular curve over k of genus>0, and let D be an ample invertible

sheaf of degree d>0. The P1-bundle Z=P(_??_Y _??_D)•¨ƒÎY Y has sections S and

T such that S•¿T=ƒÓ and the self-intersection numbers of S and T are -d

and d respectively. Let V be the open subscheme Z-S_??_iZ. Then: 1)

There is an isomorphism i*ƒÎ*: Pic Y•¨•`Pic V=Pic V(D•É). 2) For an

arbitrary invertible sheaf M on Y, i*ƒÎ*M is ample if and only if deg M>0.

3) For an arbitrary invertible sheaf M on Y of positive degree, (V, i*ƒÎ*M)

is semi-complete if and only if there are positive integers a and b such that

M_??_a_??_D_??_b. Thus taking an invertible sheaf E on Y of degree 0 and of in

fi nite order, we see that (V, i*ƒÎ*D) is semi-complete but (V, i*ƒÎ*(D_??_E)) is

not semi-complete.

PROOF. (1) is well-known (for example, see [6, (1.5)]). If i*ƒÎ*M is ample

(M•¸Pic Y), we see that M is ample by restricting i*ƒÎ*M to T. Hence the only

if part of (2) is obvious. By Nakai's criterion for ampleness, Dy(S) Ox n*M_??_2d

is ample if deg M>0. Thus the if part of (2) is proved. A direct calcula

tion shows that, for sufficient large n, _??_z (nS) _??_ƒÎ D_??_n induces a morphism

ƒÏ of Z to a normal surface W such that p contracts S to a closed point P

of W and induces an isomorphism of Z-S to W-{R}. Hence the if part

of (3) is proved. If (V, i*ƒÎ*M) is semi-complete (Me Pic Y, deg M>0), there

are a natural number n and an embedding ƒÓ: V•¨PNk with the properties

stated in Definition 1.1. Then ƒÓ can be expressed as a composite morphism

V_??_iZ•¨fPNk (Zariski's main theorem). Hence f*_??_PNk(1)_??_*M_??_n _??_ _??_z(mS)

for some integer m. Taking the inverse images under the morphism S•¨Z,

we obtain _??_Y_??_M_??_n_??_ D_??_(-m). Thus the only if part of (3) is proved because

deg M>0 and deg D>0. The existence of the required E in the last asser

tion is well-known in the theory of abelian varieties. q.e.d.

•˜ 2. Geometric graded domains

In this section, we give a criterion for a graded domain to be associated

with some semi-complete polarized variety.

DEFINITION 2.1. Let R be a graded k-algebra. We say that R is

almost-geometric if (i) R is finitely generated over k, (ii) R0 is a finite ex

tension field of k, (iii) Rn=0 for every negative integer n, and (iv) G.C.M.

{n•¸Z|Rn•‚0}=1. We say that R is geometric if there is a semi-complete

polarized variety (X, L) over k such that R_??_R(X, L).

It is clear that R is almost-geometric if R is geometric. In fact, we

have only to check that G.C.M. {n•¸Z|Rn•‚0}=1 (Theorem 1.2). This is


obvious because Lon is generated by H0(X, L®n) for sufficiently large n .

From now on until the end of this section, we fix an almost-geometric

graded domain R over k.

We understand by a minimal set of generators of R a minimal basis of

R+ as an R-module consisting of homogeneous elements. Let {u0, •c, uN} be

one of such sets. Then {u0, •c, uN} generates R as an R0-algebra [2, Lemma

(2.1.3)]. It is clear that (deg u0, •c, deg uN) is uniquely determined by R up

to a permutation, i.e. independent of the choice of {u0, •c, uN}. In the

following, we find a good open subset of Proj R with the method of our

paper [7, •˜1].

DEFINITION 2.2. With the above notation, we set m(R)=L.C.M. {deg u0,

•c, deg uN}. For an arbitrary natural number a, Ia denotes the ideal of R

generated by {ui| deg ui_??_0 mod a}, namely Ia=a_??_ n RRn. We denote by

G. Proj R the open subscheme Proj R-•¾1<a V+(Ia), and by g(R) min. ht Ia

=codimp roj R (Proj R-G. Proj R).

REMARK 2.3. G. Proj R is well-defined because V+(Ia)=~f . unless a is a

divisor of m(R). On the other hand, Ia•‚0 for every integer a>1 because

G.C.M. {neZIRn•‚0}=1. Thus we see that dim R>g(R)>1 and G . Proj R•‚

5.

THEOREM 2.4. G. Proj R is the largest among the open subsets U of

Proj R with the following two properties:

i) t(1)IU is an invertible sheaf on U.

ii) For every positive integer a, the natural homomorphism (C~(1)IU)Oxa

•¨ 9(a) is an isomorphism.

Furthermore, if g(R)>1, G. Proj R is the largest among the open sub

sets U with the property (i).

The proof is similar to that of [7, Theorem 1.7], and we omit it .

PROPOSITION 2.5. The natural homoorphism

is the Gm-bundle associated with V(~(1)IG.Proj R)

. The proof is similar to that of [7, Theorem 2.3, (1)], and we omit it.

COROLLARY 2.6. Let R be an almost-geometric graded domain over a fi eld k such that dim R>2. Then R is geometric if and only if g(R) _>2 and depth R1+>2.

230 SHIGEFUMI MORI

PROOF. Assume that R°vR(X, L) for some semi-complete polarized

variety (X, L) over k. By Corollary 1.7, depth RR+>2. By Proposition 1.5,

we see that X is embedded in Proj R. For sufficiently large n, Lon is gen

erated by Rn, i.e. X is disjoint from V+(RRn). Hence X•¿V+(Ia)=ƒÓ for

every a>1. In other words, we have Xc=~G. Proj R. Thus we see that

g(R)>2 by Corollary 1.8. Conversely, let us assume that g(R)>2 and

depth RR+>2. X=G. Proj R and L =C~(1)|x make a semi-complete polarized

variety (X, L) (Theorem 2.4). We have only to show that the natural k

algebra homomorphism f: R=ƒ¡(SpecR, (9SpecR)->ƒ¡(U, cU)=R(X, L) is bijec

tive, where U=Spec R-•¾1<a V(Ia) (cf. Corollary 1.4 and Proposition 2.5).

f is bijective because depthY (9Spec R>2 by the assumption, where Y=

•¾1<a V (Ia) (cf. [3, •˜2 and •˜3]). q.e.d.

The proof of Corollary 2.6 shows that, if R is geometric, (G. Proj R,

&(1)|G .Proj R) is the largest among the semi-complete polarized k-varieties

(X, L) such that R~R(X, L). Thus we define

DEFINITION 2.7. If R is a geometric graded domain over k , (G. Proj R,

&(1) |G .ProjR) is called the maximal associated polarized k-variety of R.

By Remark 1.11 and by the definition of g(R), we have the following:

COROLLARY 2.8. Assume that R is geometric, and let (X, L) be the as

sociated polarized k-variety of R. Then, for an arbitrary compactification

X•¨X stated in Definition 1.1, we have codimx (X-X)=g(R).

If R is factorial and of dimension~2, then depth RR+>2, every ui is a

prime element, and ht (ui, uj)=2 if i•‚j. Hence we obtain

COROLLARY 2.9. Let R be an almost-geometric graded factorial domain

over k such that dim R>2. Then R is geometric if and only if

#{i|deg ui_??_0 mod a}>2

for every integer a>1.•˜

3. The geometric parts of almost-geometric graded factorial domains

The main theorem of this section associates a semi-complete polarized

variety over k with an arbitrary almost-geometric graded factorial domain

over k. To be precise:

THEOREM 3.1. Let R be an almost-geometric graded factorial domain

over k. Assume that dim R>2. Then there is a unique natural number m


with the following two properties:i) R(m) is a geometric graded factorial domain ,

ii) Cl R(n)~Z/(n') for an arbitrary natural number n , where n'=n/(n, m).

Let us note an easy fact: R(n)°~R[n] as R0-algebras for every natural number n. This fact will be freely used later .

REMARK 3.2. Under the notation as above, assume that dim R=1 . Then RR0[X] (deg X= 1). Hence C1 R(n)=0 for every natural number n . Thus the above assumption dim R>2 is essential.

Before proving this, we give a corollary.

COROLLARY 3.3. Let n be an arbitrary natural number . Then R(n) is factorial (or, geometric) if and only if n|m (or, m|n, resp.).

PROOF. In view of the theorem above, we have only to show that m|n if R(n) is geometric. If R(c) is geometric, we see that|C1 R(mn)|=m|Cl R(n)

| by Theorem 1.9 (or by Remark 3.6, (2)). Thus we obtain n=mn/(m, n) by the property (ii) of m. This implies that m|n, q .e.d.

This corollary justifies the following definition:

DEFINITION 3.4. Let R be an almost-geometric graded factorial domain over k. If dim R>2 (or, dim P=1), we define the index of R to be the number m given in Theorem 3.1 (or, 1, resp.), the geometric part of R to be R[m] (or, R, resp.), and the associated geometric graded domain of R to be R(m) (or, R, resp.).

We prepare an important lemma.

LEMMA 3.5. Let A be an almost-geometric graded normal domain over k. For an arbitrary natural number a, we have an exact sequence

where a is the natural homomorphism C1 A[a]•¨C1 A, and eP/PfA[a](•¸N) is

the ramification index of P over P•¿A[a].

PROOF. We recall that the divisor class group Cl B of a graded Krull

domain B is isomorphic to DH B/FH B, where DH B denotes the group of

homogeneous divisors of B of height 1 and FH B denotes the subgroup of

DH B generated by principal ideals [8, Proposition 7 .1]. On the other hand,

FH B is isomorphic to QH(B)*/B*0. Thus we have the following commuta

tive diagram with exact rows:

232 SHIGEFUMI MORI

By applying the "Snake lemma" to the above diagram, we obtain the required

exact sequence. q .e.d.

For convenience, we recall, with the notation of Lemma 3.5:

REMARK 3.6. 1) Let P be a homogeneous prime ideal of height 1 of A .

Since P=•ãA(P (1 A[ƒ¿]), we have

[Q(A/P):Q((A/P)[a])]eP/PfA[a]=a,

by virtue of the theory of ramification (cf. [1, Chapitre VI, •˜8, n•‹5,

Theoreme 2]). For a natural number b, we have

eP/PnA[ab]=eP/PnA[a]ePnA[a]/PnA[ab].

2) If A is geometric and dim A~2, then eP/RnA[a]=1 for every homo

geneous prime ideal of height 1 of A. Indeed,

0•¨Z/(a)•¨ClA[a]•¨Cl A•¨0

is exact by Theorem 1.9.

The proof of Theorem 3.1 is divided into several steps. First, with

the notation and the assumption of Theorem 3.1, we have

LEMMA 3.7. There is a natural number m with the property (i).

PROOF. We prove the lemma by induction on m(R) (Definition 2.2). If

m(R)=1, then R is generated by R1 as an R0-algebra. Hence R is geometric.

In this case, m=1 enjoys the required property by virtue of Remark 3.6, (2).

Let us observe the general case. Let {u0, ... , uN} be a minimal set of gen

erators of R (cf. •˜2). By virtue of Corollary 2.9, there are a natural number

c and a prime number p such that 0`c<N, deg uc•ß0 mod p, and deg ui

•ß0 mod p for every i•‚c. Then a monomial ua00 ... uaNN (a0, ... , aN>0) is of

degree•ß0 mod p, if and only if ac•ß0 mod p. Thus R(p) is generated by up

and {ui|i•‚c} as an R0-algebra. Hence, we have m(R(p))<m(R)/p. On the

other hand, eu0R/u0RnR[p]=p because R/u0R=(R/u0R)[p] (cf. Remark 3.6, (1)).

Thus R(p) is also factorial (Lemma 3.5). By the induction hypothesis, there

is a natural number m' such that (R(p))(m')=R(pm') is factorial and geometric .

q.e.d..


In order to treat the property (ii), we need a lemma.

LEMMA 3.8. Let A be an almost-geometric graded domain over k. Let a and b be natural numbers. If A and A(ab) are factorial, so is A(a).

PROOF. As is easily seen, we have only to treat the following two cases: Case (1) a and b are relatively prime with each other, and Case (2) a is a prime number and b is a power of a.

Case (1). Consider a:Cl A[a]Cl A and j9: Cl A[ab] A°Cl (A(a))[b]Cl A(a) Cl A[a]. By virtue of Lemma 3.5 and Remark 3.6, (1), we see that every

prime divisor of Ker a (or, I Coker R ~) is a divisor of a (or, b, resp.). By the hypothesis, we have Cl A Ker a Coker ~. Hence Cl A (a)=0.

Case (2). Let us consider the homomorphism Cl A[ab]Cl A. Since ab is a power of a prime, there is a unique homogeneous prime ideal P of height 1 of A such that e P/PnA[ab]>1 (Lemma 3.5). Hence eQ/QnA[ab]=1 for every homogeneous prime ideal Q of height 1 of A[a] other than PnA[a] (Remark 3.6, (1)). Thus, in view of Remark 3.6, (1), we see that 0=Cl (A(a))[b]Cl A(a) is surj ective, and Cl A(a)=0 (Lemma 3.5), q.e.d.

The following lemma is the final step of our proof of Theorem 3.1.

LEMMA 3.9. The number m given in Lemma 3.7 enjoys the property

(ii) stated in Theorem 3.1.

PROOF. Let n be an arbitrary natural number. We set d=(m, n), n'=n/d, and m'=m/d. By Lemma 3.8, A=R(d) is factorial. Let us consider the ramification index of an arbitrary homogeneous prime ideal P of height 1 of A over P(1A[m'n']. Since A (m')=R(m) is geometric, ePflA[m']/PnA[m'n']=1 (Remark 3.6, (2)). Thus eP/PfA[m'n'] is a divisor of m', hence eP/PfA[n'] is

a divisor of m' and n' (Remark 3.6, (1)). This implies that eP/PnA[n']=1. By applying Lemma 3.5 to Cl A[n']Cl A, we see that C1 R(n)=Cl A(n')"Z/(n').

q.e.d.

Now, the uniqueness of the required natural number m is obvious by the property (ii).

•˜ 4. The construction of almost-geometric graded factorial domains

The purpose of this section is to construct arbitrary almost-geometric

graded factorial domains from geometric graded factorial domains. Since

we introduce various gradation on a graded ring and on its subrings, we

specify the gradation in expressing the degree of an element a of a graded

ring R. For example, degR (q) a=n if aeRm-{0} and nq=m, where m, neN.

234 SHIGEFUMI MORI

First, we consider a special type of ring extensions.

I. Let S be an almost-geometric graded factorial domain over k. As

sume that r(>0) natural numbers e1, ..., er (e1>...>er>1) and r homo

geneous elements v1, •c, vr of S satisfy the following conditions:

i) (ei, degs vi)=1 for arbitrary i,

ii) (ei, ej)=1 for arbitrary i and j such that i•‚j,

iii) Sv1,..., Svr are mutually distinct prune ideals of height 1.

Then we call the pair (v, e) a ramification data (of size r) over S, where v=

(v1, •c, vr) and e=(e1, ..., er). For the pair (v, e), we define a k-algebra

S[v1/e] by:

(4.1.1) S[v1/e]=S(1/m)[X1, ..., Xr]/(Xe11-v1, ..., Xerr-vr),

where m=[Iri=1 ei. S[v1/e] admits one and only one structure of a graded

S(1/m)-algebra. Indeed, we have only to define degs[v1/e] Xi=(m/ei) degs vi

(i=1, •c, r). We consider S[vi/e] as a graded k-algebra in this way.

THEOREM 4.1. R=S[v1/e] is an almost-geometric graded factorial do

main over k such that R(m)=S. RXi is a homogeneous prime ideal of R of

height 1 such that RXi(1S=Svi and eRXi/Svi=ei (i=1, •c, r). There are no

other homogeneous prime ideals P of R of height 1 such that eP/P•¿s>1.

REMARK 4.2. If S is geometric and dim S>2, then m is the index of R

(Corollary 3.3).

PROOF. Let us prove our assertion by induction on r. If r=0, the as

sertion is obvious. We observe the general case admitting that our asser

tion is true in the case r=1. We consider the graded ring T=S[(vr)1,(er)]

associated with the ramification data ((vr), (er)) over S. By our assumption,

T is an almost-geometric graded factorial domain over k such that T(er)=S.

TXr is a prime ideal such that TXr(1S=Svr and eTxr/Svr=er. If Pi is the

prime ideal of T lying over Svi (i=1, •c, r-1), then ePi/Svi=1 by our as

sumption. This implies that Pi=Tvi (i=1, •c, r-1). Hence

R=T [(v1, •c, vr-1)1/(e1,•c,er-1)],

and our assertion is easily proved by the induction hypothesis (cf. Remark

3.6, (1)). Thus we may assume that r=1. Similarly, we may assume that

e1 is a prime number. We use e, v, and X instead of e1, v1, and X1, respec

tively. Since Xe-v is a prime element of S[X] by Eisenstein's criterion

for irreducibility, we see easily that R is an almost-geometric graded domain

over c such that R(e)=S. We claim that R is normal. Since the integral

closure R of R in Q(R) is a graded subring of QH(R) [1, Chapitre V, •˜1,

n•‹9, Proposition 21], R is generated as an R0-module by the homogeneous


elements ƒÆ=sXi of R such that seQH(S) and 0CiCe . Then ƒÆe=sevieS

because S is normal. Thus seS because S is factorial . Hence R=R, and

R is normal. Since R/RX =S/Sv=(R/RX)[e], we see that RX is a prime

ideal of height 1 such that RX(1S=Sv and eRX/sv=e. Let P be an arbitrary

homogeneous prime ideal of R of height 1 such that eP/Pns>1. Since e is

prime, we have R/P=(R/P)[e] by Remark 3.6, (1). Hence xeP because

deg X0 mod e. Thus P=RX, and hence R is factorial by Lemma 3.5.

q.e.d.

II. We determine the algebra structure of an arbitrary almost

geometric graded factorial domain over its geometric part.

Let R be an arbitrary almost-geometric graded factorial domain over k

of index m (cf. Definition 3.4). Let us denote R(m) by S . Then S(1/m) is the

geometric part R[m] of R. By Lemma 3.5, there are a finite number of

homogeneous elements u1, ... , ur of R such that

{Ru1, ..., Rur}={PeProj R|ht P=1, eP/PnS>1},

Ru1•‚Rug if i•‚j, and el>...>e r>1, where ei=eRui/Ruins (i=1, ..., r). The

natural numbers e1,..., er are called the ramification indices of R. Then

vi=ueii is a prime element of S (i=1,..., r), because every generator wi(eS)

of Rui(1S is expressed in R as a product wi=ciueii, where cieR*0. Then:

LEMMA 4.3. m=ffri =1 ei, and (v, e) is a ramification data over S, where

v=(v1,•c,vr) and e=(e1, ..., er).

PROOF. By Lemma 3.5, we see that Z/(m)^'fri=1 Z/(ei). Hence m=

fri=1 ei and (ei, ej)=l if i•‚j . Let b be ei/(ej, degs v2). Then ubiS(1/m) be

cause degR ubi=(bm/ ei) degs vi=0 mod m . Hence b=0 mod ei by the defini

tion of ramification index ei. Thus (ei, degs vi)=1 for every i. It is obvious

that Svi•‚Svj if i•‚j. q .e.d.

Now, we have:

THEOREM 4.4. An arbitrary almost-geometric graded factorial domain

R over k is obtained as S[vl/e], where S is the associated geometric graded

domain of R (Definition 3.4), and (v, e) is a ramification data over S .

PROOF. Let us maintain the notation of Lemma 4 .3. We assert that

the natural homomorphism ~b: S[vl/e]R, such that 2iXi=ui for every i, is

an isomorphism. We prove it by induction on r. If r=0, our assertion is

obvious. Let us observe the general case . By applying the induction as

sumption to R(er), we have the natural isomorphism:

236 SHIGEFUMI MORI

Thus our assertion is reduced to the bijectivity of the following natural homomorphism:

where ƒµXr=ur. Since eRur/RurfR[er]=er by Remark 3.6, (1), we see that

R/Rur=(R/Rur) [er], and R=Rur+R[er] (Remark 3.6, (1)). Hence R=R[er] [ur],

and ƒµ is surjective. Since R[er][Xr]/(Xerr-vr)"'(R[er])oer as R[er]-modules,

Ker ƒµ is a torsionfree R[er]-module. Thus Ker ƒµ=0 because ƒµ Ox Q(R[er]) is

obviously an isomorphism. q.e.d.

III. We classify almost-geometric graded factorial domains by geo

metric graded factorial domains and ramification data.

Let S (or, T) be a geometric graded factorial domain over k of dimen

sion>2, (v, e) (or, (w, f )) a ramification data over S (or, T, resp.), where v=

(v1, ..., Vr), e=(e1, ..., er), and w=(w1, ..., we). Then:

THEOREM 4.5. 1) S[v1/e]^'T [w1/f] as graded k-algebras if and only if

e=f (hence r=s) and there is an isomorphism ƒÓ:T•`S of graded k

algebras such that S[vl/e]v1=S[vl/e]ƒÓwi (i=1, ..., r).

2) Assume that S=T and e=f. Then S[vl/e]~'S[wl/f] as graded S(1/m)-

algebras (m=fjri=1 ei) if and only if there are elements c1, •c, cr of So such

that vi= ceiiwi (i=1, •c,r).

3) Let (X, L) be the maximal associated polarized variety of S, Yi the

closed subset of X defined by vi (i=1, ..., r). Then we have an exact

sequence of groups :

where Autgr . k-alg. S[vl/e] denotes the automorphism group of the graded k

algebra S[vl/e], and Autk (X; Y1, ..., Yr) denotes the group of automorphisms

g of the k-scheme X such that gYi=Yi (i=1, ..., r).

PROOF. The assertion (2) is obvious. Let us prove (3), ƒ¿ is defined by

ƒ¿(c)(r) = cir (c•¸S*0, i•¸Z, and r•¸S[vl/e]i). Every element D of Autgr. k-alg. S[v1/e]

induces an element ƒÓ of Autgr . k-alg. S Such that SƒÓvi=Svi for every i (Theorem

4.1 and Remark 4.2). Thus we define fi(b) to be the element of Autk (X; Y1,

..., Yr) induced by c-1. It is obvious that j3 is a homomorphism, Ker a={1},

and Im a= Ker 43. It remains to show that 48 is surjective. Let g be an

arbitrary element of Autk (X; Y1, ..., Yr). Then g*L^dL by Corollary 1.10.

Hence g-1 is induced by some element ƒÓ of Autgr . k-alg. S. Since gYi=Yi for

every i, ci=vi/ƒÓvi•¸S*0 for every i. We claim that there exist elements x,

y1,..., yr of S*0 such that ci=xdiyeii for every i, where di=degs vi. Indeed,


the required elements exist because (ei, di)=1 for every i, and (ei, ej) =1 if i •‚j

(by the definition of ramification data). Let be the element of Autgr. k-alg, S

such that ƒµa=xia (i•¸Z and aƒµSi). Then vi=peiiƒµƒÓvi for every i. Hence

we obtain an automorphism b e Autgr . k-alg. S[vl/e] by defining oa=ƒµƒÓa for

every a•¸S and TbXi=yi-1Xi for every i (cf. (4.1.1)). Then j3(~)=g, and hence

(3) is proved. The only if part of (1) is due to Theorem 4.1 and Remark 4.2.

The if part of (1) is proved with the method of the latter half of the proof

of (3), and we omit it. q.e.d.

•˜ 5. An application

Let us assume that k is algebraically closed. We classify almost-geo

metric graded factorial domains over k of dimension 2.

Let R be an arbitrary almost-geometric graded factorial domain over

k of dimension 2 with r(>0) ramification indices ei, ..., er. (e1> ... er>1).

Then we have, necessarily, by Lemma 4.3:

(ei, ej)=1 for arbitrary i and j such that i•‚j.

Under these notation and assumption, our classification is stated as follows:

THEOREM 5.1. a) If r<2, then

Rk[X1, X2], where deg Xi=fle1 (i=1,2).

b) If v>3, then the set

Mr=(a3, ..., ar)•¸(k*)r-2|a3=1, ai=aj if i=j}

is in one to one correspondence T with the set F(r; e1, ..., er) of isomor

phism classes of such graded k-algebras R. T maps (a3, ..., ar)•¸Mr to the

class ofT(a3, ..

., ar)=k[X1, ..., Xr]/(Xe11+a3Xe22+Xe33, ..., Xe11+are22+Xerr),

where deg Xi=(1Arj= ej)/ei (i=1, ..., r).

REMARK 5.2. Theorem 5.1 shows that every minimal set of generators

of R consists of max. {r, 2} elements.

REMARK 5.3. The factoriality of T(1) given in the above theorem was

fi rst proved by P. Samuel [8, p. 32].

PROOF. Let S be an arbitrary geometric graded factorial domain over

k of dimension 2. Let (X, L) be the maximal associated polarized k-variety

238 S HIGEFUMI MORI

of S. Then X is a complete non-singular curve over k such that Pic X Z

hence X is of genus 0. Hence X P1k, and L (9P1k(l) because cl(L) generates

Pic X. Thus we have S~k[Z1, Z2], where deg Z1=deg Z2=1. Since k is alge

braically closed, every homogeneous prime element (S0) of S is of degree

1. Since we have Autgr . k-alg. S ~°Autk-mod. S1GL2 (k), we can normalize the ramification data over S by Theorem 4.5, (1). Namely, let (v, e) be an

arbitrary ramification data over S such that e =(e1, ..., er). If r<2, then

S[vl/e],vS[wl/e], where w=(), (Z1), or (Z1, Z2). If r>3, there is one and only

one point (a3, ..., ar) of Mr such that S[vl/e] S[wl/e] as graded k-algebras,

where w=(Z1, Z2, -Z1-a3Z2, ..., -Z1-arZ2). Now, our assertion follows

immediately from Theorem 4.4. q.e.d.

References

[1] N. Bourbaki, Elements de mathematique, Algebre commutative, Chapitres 5 et 6, Hermann, Paris, 1964.

[2] A. Grothendieck and J. Dieudonne, Elements de geometrie algebrique, Chapitre II, Inst. Hautes Etudes Sci. Publ. Math., 8.

[3] A. Grothendieck, Local cohomology, Lecture Notes in Mathematics, No. 41, Springer, Berlin, 1967.

[4] A. Grothendieck, Seminaire de geometrie algebrique, 1, Springer Lecture Notes, No. 224.

[5] A. Grothendieck and N. H. Kuiper, Cohomologie locale des faisceaux coherents et theoremes de Lefschetz locaux et globaux (SGA 2), North-Holland Publ. Comp.

Amsterdam-Paris, 1962.

[6] S. Mori, On affiine cones associated with polarized varieties, Japan. J. Math., 1(2), 1975.

[7] S. Mori, On a generalization of complete intersections, J. Math. Kyoto Univ., 15(3), 1975.

[8] P. Samuel, Lectures on unique factorization domains, Tata Inst. Fund. Ides., Bombay, 1964.

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