graded factorial domains - j-stage
TRANSCRIPT
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 ĵ
: 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]).
Graded factorial domains 227
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-ĵ(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
Graded factorial domains 229
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
Graded factorial domains 231
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..
Graded factorial domains 233
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
Graded factorial domains 235
elements Į=sXi of R such that seQH(S) and 0CiCe . Then Įe=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 ĵXr=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 ĵ is surjective. Since R[er][Xr]/(Xerr-vr)"'(R[er])oer as R[er]-modules,
Ker ĵ is a torsionfree R[er]-module. Thus Ker ĵ=0 because ĵ 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,
Graded factorial domains 237
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.
DEPARTMENT OF MATHEMATICS
KYOTO UNIVERSITY
KYOTO 606 JAPAN