MIMS Technical Report No.00009 (200804141)

MULTIGRADED RINGS, DIAGONAL SUBALGEBRAS, AND
RATIONAL SINGULARITIESKAZUHIKO KURANO, EI-ICHI SATO, ANURAG K. SINGH, AND KEI-ICHI WATANABE

1. Introduction
We study the properties of F-rationality and F-regularity in multigraded rings
and their diagonal subalgebras. The main focus is on diagonal subalgebras of bigraded rings: these constitute an interesting class of rings since they arise naturally
as homogeneous coordinate rings of blow-ups of projective varieties.
Let X be a projective variety over a field K, with homogeneous coordinate ring A.
Let a ⊂ A be a homogeneous ideal, and V ⊂ X the closed subvariety defined by a.
For g an integer, we use ag to denote the K-vector space consisting of homogeneous
elements of a of degree g. If g ≫ 0, then ag defines a very ample complete linear
system on the blow-up of X along V , and hence K[ag ] is a homogeneous coordinate
ring for this blow-up. Since the ideals ah define the same subvariety V , the rings
K[(ah )g ] are homogeneous coordinate ring for the blow-up provided g ≫ h > 0.
Suppose that A is a standard N-graded K-algebra, and consider the N2 -gradingon the Rees algebra A[at], where deg rtj = (i, j) for r ∈ Ai . The connection withdiagonal subalgebras stems from the fact that if ah is generated by elements ofdegree less than or equal to g, thenK[(ah )g ] ∼A[at](gk,hk) .
=k0Using ∆ = (g, h)Z to denote the (g, h)-diagonal in Z2 , the diagonal subalgebraA[at]∆ = ⊕k A[at](gk,hk) is a homogeneous coordinate ring for the blow-up of Proj A
along the subvariety defined by a, whenever g ≫ h > 0.
The papers [GG, GGH, GGP, Tr] use diagonal subalgebras in studying blowups of projective space at finite sets of points. For A a polynomial ring and aa homogeneous ideal, the ring theoretic properties of K[ag ] are studied by Simis,
Trung, and Valla in [STV] by realizing K[ag ] as a diagonal subalgebra of the Rees
algebra A[at]. In particular, they determine when K[ag ] is Cohen-Macaulay for a acomplete intersection ideal generated by forms of equal degree, and also for a theDate: March 24, 2008.
2000 Mathematics Subject Classification. Primary 13A02; Secondary 13A35, 13H10, 14B15.
A.S. was supported in part by grants from the National Science Foundation.
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KAZUHIKO KURANO, EI-ICHI SATO, ANURAG K. SINGH, AND KEI-ICHI WATANABE

ideal of maximal minors of a generic matrix. Some of their results are extended byConca, Herzog, Trung, and Valla as in the following theorem:
Theorem 1.1. [CHTV, Theorem 4.6] Let K[x1 , . . . , xm ] be a polynomial ring overa field, and let a be a complete intersection ideal minimally generated by forms ofdegrees d1 , . . . , dr . Fix positive integers g and h with g/h > d = max{d1 , . . . , dr }.
Pr
Then K[(ah )g ] is Cohen-Macaulay if and only if g > (h − 1)d − m + j=1 dj .
When A is a polynomial ring and a an ideal for which A[at] is Cohen-Macaulay,
Lavila-Vidal [Lv1, Theorem 4.5] proved that the diagonal subalgebras K[(ah )g ] are
Cohen-Macaulay for g ≫ h ≫ 0, thereby settling a conjecture from [CHTV]. In[CH] Cutkosky and Herzog obtain affirmative answers regarding the existence ofa constant c such that K[(ah )g ] is Cohen-Macaulay whenever g > ch. For morework on the Cohen-Macaulay and Gorenstein properties of diagonal subalgebras,see [HHR, Hy2, Lv2], and [LvZ].
As a motivating example for some of the results of this paper, consider a polynomial ring A = K[x1 , . . . , xm ] and an ideal a = (z1 , z2 ) generated by relatively
prime forms z1 and z2 of degree d. Setting ∆ = (d + 1, 1)Z, the diagonal subalgebraA[at]∆ is a homogeneous coordinate ring for the blow-up of Proj A = Pm−1 along
the subvariety defined by a. The Rees algebra A[at] has a presentation
R = K[x1 , . . . , xm , y1 , y2 ]/(y2 z1 − y1 z2 ) ,where deg xi = (1, 0) and deg yj = (d, 1), and consequently R∆ is the subalgebra of
R generated by the elements xi yj . When K has characteristic zero and z1 and z2
are general forms of degree d, the results of Section 3 imply that R∆ has rational
singularities if and only if d 6 m, and that it is of F-regular type if and only
if d < m. As a consequence, we obtain large families of rings of the form R∆ ,standard graded over a field, which have rational singularities, but which are notof F-regular type.
It is worth pointing out that if R is an N2 -graded ring over an infinite fieldR(0,0) = K, and ∆ = (g, h)Z for coprime positive integers g and h, then R∆ is thering of invariants of the torus K ∗ acting on R via
λ : r 7−→ λhi−gj r

where λ ∈ K ∗ and r ∈ R(i,j) .

Consequently there exist torus actions on hypersurfaces for which the rings of invariants have rational singularities but are not of F-regular type.
In Section 4 we use diagonal subalgebras to construct standard graded normal
2
2
rings R, with isolated singularities, for which Hm(R)0 = 0 and Hm(R)1 6= 0. If
S is the localization of such a ring R at its homogeneous maximal ideal, then,by Danilov’s results, the divisor class group of S is a finitely generated abeliangroup, though S does not have a discrete divisor class group. Such rings R are
also of interest in view of the results of [RSS], where it is proved that the image ofMULTIGRADED RINGS

3

2
2
Hm
(R)0 in Hm(R+ ) is annihilated by elements of R+ of arbitrarily small positive2
degree; a corresponding result for Hm(R)1 is not known at this point, and the rings
constructed in Section 4 constitute interesting test cases.
Section 2 summarizes some notation and conventions for multigraded rings andmodules. In Section 3 we carry out an analysis of diagonal subalgebras of bigradedhypersurfaces; this uses results on rational singularities and F-regular rings provedin Sections 5 and 6 respectively.
The authors would like to thank Shiro Goto and Ken-ichi Yoshida for theirvaluable comments.

2. Preliminaries
In this section, we provide a brief treatment of multigraded rings and modules;see [GW1, GW2, HHR], and [HIO] for further details.
By an Nr -graded ring we mean a ring
M
R=
Rn ,
n∈Nr

which is finitely generated over the subring R0 . If (R0 , m) is a local ring, then R
has a unique homogeneous maximal ideal M = mR + R+ , where R+ = ⊕n6=0 Rn .
For m = (m1 , . . . , mr ) and n = (n1 , . . . , nr ) in Zr , we say n > m (resp. n > m)if ni > mi (resp. ni > mi ) for each i.
Let M be a Zr -graded R-module. For m ∈ Zr , we set
M
M>m =
Mn ,
n>m

which is a Zr -graded submodule of M . One writes M (m) for the Zr -graded Rmodule with shifted grading [M (m)]n = Mm+n for each n ∈ Zr .
Let M and N be Zr -graded R-modules. Then HomR (M, N ) is the Zr -graded
module with [HomR (M, N )]n being the abelian group consisting of degree preserving R-linear homomorphisms from M to N (n).
The functor ExtiR (M, −) is the i-th derived functor of HomR (M, −) in the category of Zr -graded R-modules. When M is finitely generated, ExtiR (M, N ) and
ExtiR (M, N ) agree as underlying R-modules. For a homogeneous ideal a of R, thelocal cohomology modules of M with support in a are the Zr -graded modules
Hai (M ) = lim ExtiR (R/an , M ) .
−→n
Let ϕ : Zr −→ Zs be a homomorphism of abelian groups satisfying ϕ(Nr ) ⊆ Ns .
We write Rϕ for the ring R with the Ns -grading whereM
Rm .
[Rϕ ]n =ϕ(m)=n4

KAZUHIKO KURANO, EI-ICHI SATO, ANURAG K. SINGH, AND KEI-ICHI WATANABE

If M is a Zr -graded R-module, then M ϕ is the Zs -graded Rϕ -module withM
Mm .
[M ϕ ]n =ϕ(m)=nϕ

The change of grading functor (−) is exact; by [HHR, Lemma 1.1] one hasiiϕHM(M )ϕ = HM
ϕ (M ) .

Consider the projections ϕi : Zr −→ Z with ϕi (m1 , . . . , mr ) = mi , and set dim R(R)ϕi a 6= 0 ;a(Rϕi ) = max a ∈ Z | HM
this is the a-invariant of the N-graded ring Rϕi in the sense of Goto and Watanabe[GW1]. As in [HHR], the multigraded a-invariant of R is
a(R) = a(Rϕ1 ), . . . , a(Rϕr ) .
Let R be a Z2 -graded ring and let g, h be positive integers. The subgroup∆ = (g, h)Z is a diagonal in Z2 , and the corresponding diagonal subalgebra of R isMR∆ =R(gk,hk) .
k∈Z

2

Similarly, if M is a Z -graded R-module, we setMM∆ =M(gk,hk) ,k∈Zwhich is a Z-graded module over the Z-graded ring R∆ .
Lemma 2.1. Let A and B be N-graded normal rings, finitely generated over a
field A0 = K = B0 . Set T = A ⊗K B. Let g and h be positive integers and set∆ = (g, h)Z. Let a, b, and m denote the homogeneous maximal ideals of A, B, andT∆ respectively. Then, for each q > 0 and i, j, k ∈ Z, one hasqT (i, j)∆ k = Ai+gk ⊗ Hbq (B)j+hk ⊕ Haq (A)i+gk ⊗ Bj+hk ⊕HmMHaq1 (A)i+gk ⊗ Hbq2 (B)j+hk .
q1 +q2 =q+1

Proof. Let A(g) and B (h) denote the respective Veronese subrings of A and B. Set
M
M
A(g,i) =Ai+gkand
B (h,j) =Bj+hk .
k∈Z

k∈Z

(g)

(h)

These are graded modules over A and B
respectively, andM
T (i, j)∆ =Ai+gk ⊗K Bj+hk = A(g,i) # B (h,j) .
k∈Z

(g)
(h)
The ideal A+ A is a-primary; likewise, B+ B is b-primary. The Künneth formula

for local cohomology, [GW1, Theorem 4.1.5], now gives the desired result.



Notation 2.2. We use bold letters to denote lists of elements, e.g., z = z1 , . . . , zs
and γ = γ1 , . . . , γs .

MULTIGRADED RINGS

5

3. Diagonal subalgebras of bigraded hypersurfaces
We prove the following theorem about diagonal subalgebras of N2 -graded hypersurfaces. The proof uses results proved later in Sections 5 and 6.
Theorem 3.1. Let K be a field, let m, n be integers with m, n > 2, and let
R = K[x1 , . . . , xm , y1 , . . . , yn ]/(f )be a normal N2 -graded hypersurface where deg xi = (1, 0), deg yj = (0, 1), anddeg f = (d, e) > (0, 0). For positive integers g and h, set ∆ = (g, h)Z. Then:(1) The ring R∆ is Cohen-Macaulay if and only if ⌊(d − m)/g⌋ < e/h and⌊(e − n)/h⌋ < d/g. In particular, if d < m and e < n, then R∆ is CohenMacaulay for each diagonal ∆.
(2) The graded canonical module of R∆ is R(d − m, e − n)∆ . Hence R∆ is
Gorenstein if and only if (d − m)/g = (e − n)/h, and this is an integer.
If K has characteristic zero, and f is a generic polynomial of degree (d, e), then:(3) The ring R∆ has rational singularities if and only if it is Cohen-Macaulay
and d < m or e < n.
(4) The ring R∆ is of F-regular type if and only if d < m and e < n.
For m, n > 3 and ∆ = (1, 1)Z, the properties of R∆ , as determined by m, n, d, e,
are summarized in Figure 1.
Gorenstein: e=d−m+n0000000000000000000000
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00 Cohen−Macaulay
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00 Rational Singularities
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m−1

Figure 1. Properties of R∆ for ∆ = (1, 1)Z.

6

KAZUHIKO KURANO, EI-ICHI SATO, ANURAG K. SINGH, AND KEI-ICHI WATANABE

Remark 3.2. Let m, n > 2. A generic hypersurface of degree (d, e) > (0, 0) in
m, n variables is normal precisely whenm > min(2, d)

andn > min(2, e) .

Suppose that m = 2 = n, and that f is nonzero. Then dim R∆ = 2; since R∆ isgenerated over a field by elements of equal degree, R∆ is of F-regular type if andonly if it has rational singularities; see [Wa2]. This is the case precisely if
d = 1, e 6 h + 1 , ore = 1, d 6 g + 1 .
Following a suggestion of Hara, the case n = 2 and e = 1 was used in [Si, Example 7.3] to construct examples of standard graded rings with rational singularities
which are not of F-regular type.
Proof of Theorem 3.1. Set A = K[x], B = K[y], and T = A ⊗K B. By Lemma 2.1,qHm(T∆ ) = 0 for q 6= m + n − 1. The local cohomology exact sequence induced byf

0 −−−−→ T (−d, −e)∆ −−−−→ T∆ −−−−→ R∆ −−−−→ 0q
q−1
therefore gives Hm(R∆ ) = Hm
(T (−d, −e)∆ ) for q 6 m + n − 2, and also shows
m+n−2
m+n−1
that Hm(R∆ ) and Hm(R∆ ) are, respectively, the kernel and cokernel offm+n−1(T (−d, −e)∆ )
Hm

−−−−→

m+n−1
(T∆ )
Hm

f

[Ham (A(−d)) ⊗ Hbn (B(−e))]∆ −−−−→ [Ham (A) ⊗ Hbn (B)]∆ .
The horizontal map above is surjective since its graded dualf[A(d − m) ⊗ B(e − n)]∆ ←−−−− [A(−m) ⊗ B(−n)]∆T (d − m, e − n)∆

f

←−−−−

T (−m, −n)∆is injective. In particular, dim R∆ = m + n − 2.
It follows from the above discussion that R∆ is Cohen-Macaulay if and only ifq
Hm (T (−d, −e)∆ ) = 0 for each q 6 m + n − 2. By Lemma 2.1, this is the case if
and only if, for each integer k, one has
A−d+gk ⊗ Hbn (B)−e+hk = 0 = Ham (A)−d+gk ⊗ B−e+hk .
Hence R∆ is Cohen-Macaulay if and only if there is no integer k satisfying
d/g 6 k 6 (e − n)/hore/h 6 k 6 (d − m)/g ,which completes the proof of (1).
For (2), note that the graded canonical module of R∆ is the graded dual ofm+n−2Hm(R∆ ), and hence that it equalsfcoker T (−m, −n)∆ −→ T (d − m, e − n)∆ = R(d − m, e − n)∆ .

MULTIGRADED RINGS

7

This module is principal if and only if R(d − m, e − n)∆ = R∆ (a) for some integer a,i.e., d − m = ga and e − n = ha.
When f is a general polynomial of degree (d, e), the ring R∆ has an isolated
singularity. Also, R∆ is normal since it is a direct summand of the normal ring
R. By Theorem 5.1, R∆ has rational singularities precisely if it is Cohen-Macaulay
and a(R∆ ) < 0; this proves (3).
It remains to prove (4). If d < m and e < n, then Theorem 5.2 implies that R
has rational singularities. By Theorem 6.2, it follows that for almost all primes p,
the characteristic p models Rp of R are F-rational hypersurfaces which, therefore,
are F-regular. Alternatively, Rp is a generic hypersurface of degree (d, e) < (m, n),so Theorem 6.5 implies that Rp is F-regular. Since (Rp )∆ is a direct summand of
Rp , it follows that (Rp )∆ is F-regular. The rings (Rp )∆ are characteristic p modelsof R∆ , so we conclude that R∆ is of F-regular type.
Suppose R∆ has F-regular type, and let (Rp )∆ be a characteristic p model whichis F-regular. Fix an integer k > d/g. Then Proposition 6.3 implies that there exists
an integer q = pe such thatm+n−2
rankK ((Rp )∆ )k 6 rankK [Hm(ω (q) )]k ,where ω is the graded canonical module of (Rp )∆ . Using (2), we see that
m+n−2
m+n−2
Hm
(ω (q) ) = Hm
(Rp (qd − qm, qe − qn)∆ ) .

Let Tp be a characteristic p model for T such that Tp /f Tp = Rp . Multiplication
by f on Tp induces a local cohomology exact sequence
m+n−2
m+n−2
(Rp (qd − qm, qe − qn)∆ )(Tp (qd − qm, qe − qn)∆ ) −→ Hm· · · −→ Hm
p
p
m+n−1
(Tp (qd − qm − d, qe − qn − e)∆ ) −→ · · · .
−→ Hm
p
m+n−2
(Tp (qd − qm, qe − qn)∆ ) vanishes by Lemma 2.1, we conclude that
Since Hmpm+n−1
(Tp (qd − qm − d, qe − qn − e)∆ )]krankK ((Rp )∆ )k 6 rankK [Hmp= rankK Hamp (Ap )qd−qm−d+gk ⊗ Hbnq (Bp )qe−qn−e+hk .
Hence qd− qm− d+ gk < 0; as d− gk < 0, we conclude d < m. Similarly, e < n. 
We conclude this section with an example where a local cohomology module of a22
standard graded ring is not rigid in the sense that Hm(R)0 = 0 while Hm(R)1 6= 0.
Further such examples are constructed in Section 4.
Proposition 3.3. Let K be a field and let
R = K[x1 , x2 , x3 , y1 , y2 ]/(f )where deg xi = (1, 0), deg yj = (0, 1), and deg f = (d, e) for d > 4 and e > 1. Let
g and h be positive integers such that g 6 d − 3 and h > e, and set ∆ = (g, h)Z.
22
Then Hm(R∆ )0 = 0 and Hm(R∆ )1 6= 0.

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KAZUHIKO KURANO, EI-ICHI SATO, ANURAG K. SINGH, AND KEI-ICHI WATANABE

Proof. Using the resolution of R over the polynomial ring T as in the proof of
Theorem 3.1, we have an exact sequence
2
2
3
3
Hm
(T∆ ) −→ Hm(R∆ ) −→ Hm(T (−d, −e)∆ ) −→ Hm(T∆ ) .
2
3
Lemma 2.1 implies that Hm(T∆ ) = 0 = Hm(T∆ ). Hence, again by Lemma 2.1,
2
Hm(R∆ )0 = H 3 (A)−d ⊗ B−e = 0

and

2
Hm
(R∆ )1 = H 3 (A)g−d ⊗ Bh−e 6= 0 . 

4. Non-rigid local cohomology modules
We construct examples of standard graded normal rings R over C, with only22
isolated singularities, for which Hm(R)0 = 0 and Hm(R)1 6= 0. Let S be thelocalization of such a ring R at its homogeneous maximal ideal. By results ofDanilov [Da1, Da2], Theorem 4.1 below, it follows that the divisor class group of
S is finitely generated, though S does not have a discrete divisor class group, i.e.,
the natural map Cl(S) −→ Cl(S[[t]]) is not bijective. Here, remember that if A isa Noetherian normal domain, then so is A[[t]].
Theorem 4.1. Let R be a standard graded normal ring, which is finitely generated
as an algebra over R0 = C. Assume, moreover, that X = Proj R is smooth. Set(S, m) to be the local ring of R at its homogeneous maximal ideal, and Sb to be them-adic completion of S. Then
(1) the group Cl(S) is finitely generated if and only if H 1 (X, OX ) = 0;b is bijective if and only if H 1 (X, OX (i)) = 0 for(2) the map Cl(S) −→ Cl(S)each integer i > 1; and(3) the map Cl(S) −→ Cl(S[[t]]) is bijective if and only if H 1 (X, OX (i)) = 0
for each integer i > 0.
The essential point in our construction is in the following proposition:
Theorem 4.2. Let A be a Cohen-Macaulay ring of dimension d > 2, which is astandard graded algebra over a field K. For s > 2, let z1 , . . . , zs be a regular sequencein A, consisting of homogeneous elements of equal degree, say k. Consider the Rees
ring R = A[z1 t, . . . , zs t] with the Z2 -grading where deg x = (n, 0) for x ∈ An , anddeg zi t = (0, 1).
Let ∆ = (g, h)Z where g, h are positive integers, and let m denote the homogeneous maximal ideal of R∆ . Then:q(1) Hm(R∆ ) = 0 if q 6= d − s + 1, d; andd−s+1(2) Hm(R∆ )i 6= 0 if and only if 1 6 i 6 (a + ks − k)/g, where a is thea-invariant of A.

In particular, R∆ is Cohen-Macaulay if and only if g > a + ks − k.

MULTIGRADED RINGS

9

Example 4.3. For d > 3, let A = C[x0 , . . . , xd ]/(f ) be a standard graded hypersurface such that Proj A is smooth over C. Take general k-forms z1 , . . . , zd−1 ∈ A,
and consider the Rees ring R = A[z1 t, . . . , zd−1 t]. Since (z) ⊂ A is a radical ideal,gr((z), A) ∼= A/(z)[y1 , . . . , yd−1 ]is a reduced ring, and therefore R = A[z1 t, . . . , zd−1 t] is integrally closed in A[t].
Since A is normal, so is R. Note that Proj R∆ is the blow-up of Proj A at thesubvariety defined by (z), i.e., at k d−1 (deg f ) points. It follows that Proj R∆ issmooth over C. Hence R∆ is a standard graded C-algebra, which is normal andhas an isolated singularity.
If ∆ = (g, h)Z is a diagonal with 1 6 g 6 deg f + k(d − 2) − (d + 1) and h > 1,
then Theorem 4.2 implies that2
Hm(R∆ )0 = 0

and

2
Hm
(R∆ )1 6= 0 .

The rest of this section is devoted to proving Theorem 4.2. We may assume thatthe base field K is infinite. Then one can find linear forms x1 , . . . , xd−s in A suchthat x1 , . . . , xd−s , z1 , . . . , zs is a maximal A-regular sequence.
We will use the following lemma; the notation is as in Theorem 4.2.
Lemma 4.4. Let a be the homogeneous maximal ideal of A. Set I = (z1 , . . . , zs )A.
Let r be a positive integer.
(1) Haq (I r ) = 0 if q 6= d − s + 1, d.
(2) Assume d > s. Then, Had−s+1 (I r )i 6= 0 if and only if i 6 a + ks + rk − k.
(3) Assume d = s. Then, Had−s+1 (I r )i 6= 0 if and only if 0 6 i 6 a+ks+rk−k.
Proof. Recall that A and A/I r are Cohen-Macaulay rings of dimension d and d − s,respectively. By the exact sequence
0 −→ I r −→ A −→ A/I r −→ 0
we obtaindHa (A)Haq (I r ) = Had−s (A/I r )0

if q = dif q = d − s + 1
if q 6= d − s + 1, d ,which proves (1).
Next we prove (2) and (3). Since A/I r is a standard graded Cohen-Macaulay
ring of dimension d − s, it is enough to show that the a-invariant of this ring equals
a + ks + rk − k. This is straightforward if r = 1, and we proceed by induction.
Consider the exact sequence
0 −→ I r /I r+1 −→ A/I r+1 −→ A/I r −→ 0 .
Since z1 , . . . , zs is a regular sequence of k-forms, I r /I r+1 is isomorphic to(s−1+r)r(A/I)(−rk).

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KAZUHIKO KURANO, EI-ICHI SATO, ANURAG K. SINGH, AND KEI-ICHI WATANABE

Thus, we have the following exact sequence:(s−1+r)r
0 −→ Had−s (A/I)(−rk)−→ Had−s (A/I r+1 ) −→ Had−s (A/I r ) −→ 0 .
The a-invariant of (A/I)(−rk) equals a+ ks+ rk, and that of A/I r is a+ ks+ rk − kby the inductive hypothesis. Thus, A/I r+1 has a-invariant a + ks + rk.
Proof of Theorem 4.2. Let B = K[y1 , . . . , ys ] be a polynomial ring, and set
T = A ⊗K B = A[y1 , . . . , ys ] .
Consider the Z2 -grading on T where deg x = (n, 0) for x ∈ An , and deg yi = (0, 1)
for each i. One has a surjective homomorphism of graded rings
T −→ R = A[z1 t, . . . , zs t]

where yi 7−→ zi t ,and this induces an isomorphismz ... zR∼
= T /I2 ( y11 ... yss ) .

The minimal free resolution of R over T is given by the Eagon-Northcott complex
0 −→ F −(s−1) −→ F −(s−2) −→ · · · −→ F 0 −→ 0 ,where F 0 = T (0, 0), and F −i for 1 6 i 6 s − 1 is the direct sum of



s
i+1

copies ofT (−k, −i) ⊕ T (−2k, −(i − 1)) ⊕ · · · ⊕ T (−ik, −1) .
Let n be the homogeneous maximal ideal of T∆ . One has the spectral sequence:•p+q
E2p,q = H p (Hnq (F∆)) =⇒ Hm
(R∆ ) .

Let G be the set of (n, m) such that T (n, m) appears in the Eagon-Northcottcomplex above, i.e., the elements of G are(0, 0) ,(−k, −1),(−k, −2), (−2k, −1) ,(−k, −3), (−2k, −2), (−3k, −1) ,..
.
(−k, −(s − 1)),
. . . (−(s − 1)k, −1) .
Let a and b be the homogeneous maximal ideal of A and B respectively. For
integers n and m, the Künneth formula gives
H qn (T (n, m))= Hnq (A(n) ⊗K B(m))
= Haq (A(n)) ⊗ B(m) ⊕ A(n) ⊗ Hbq (B(m)) ⊕= Haq (T (n, m)) ⊕ Hbq (T (n, m)) ⊕M

i+j=q+1M

Hai (A(n)) ⊗ Hbj (B(m))

i+j=q+1

Hai (A(n)) ⊗K Hbj (B(m)) .

MULTIGRADED RINGS

11

As A and B are Cohen-Macaulay of dimension d and s respectively, it follows that
Hnq (F • ) = 0

if q 6= s, d, d + s − 1 .

In the case where d > s, one has
Hns (F • ) = Hbs (F • )andHnd (F • ) = Had (F • ) ,and if d = s, thenHnd (F • ) = Had (F • ) ⊕ Hbs (F • ) .
We claim Hbs (F • )∆ = 0. If not, there exists (n, m) ∈ G and ℓ ∈ Z such thatHbs (T (n, m))(gℓ,hℓ) 6= 0 .
This implies thatHbs (T (n, m))(gℓ,hℓ) = A(n)gℓ ⊗K Hbs (B(m))hℓ = An+gℓ ⊗K Hbs (B)m+hℓ
is nonzero, so
n + gℓ > 0andm + hℓ 6 −s ,and hence

s+m
n
6ℓ6−
.
g
h
But (n, m) ∈ G, so n 6 0 and m > −(s − 1), implying that−106ℓ6− ,hwhich is not possible. This proves that Hbs (F • )∆ = 0. Thus, we have0
if q 6= d, d + s − 1,
Hnq (F • )∆ =
H d (F • )if q = d.
a∆

It follows that•p,q
E2p,q = H p (Hnq (F∆)) = E∞for each p and q. Therefore,i•Hm(R∆ ) = E2i−d,d = H i−d (Hnd (F∆)) = H i−d (Had (F • )∆ ) = Hai (R)∆for d − s + 1 6 i 6 d − 1, andiHm(R∆ ) = 0

for i < d − s + 1 .

We next study Hai (R). Since
R = A ⊕ I(k) ⊕ I 2 (2k) ⊕ · · · ⊕ I r (rk) ⊕ · · · ,we have
Hai (R) = Hai (A) ⊕ Hai (I)(k) ⊕ Hai (I 2 )(2k) ⊕ · · · ⊕ Hai (I r )(rk) ⊕ · · · .
Theorem 4.2 (1) now follow using Lemma 4.4 (1).
Assume that d > s. Then, by Lemma 4.4 (2), Had−s+1 (I r (rk))i 6= 0 if and only
if i 6 a + ks − k.

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KAZUHIKO KURANO, EI-ICHI SATO, ANURAG K. SINGH, AND KEI-ICHI WATANABE

Assume that d = s. Then, by Lemma 4.4 (3), Had−s+1 (I r (rk))i 6= 0 if and only
if −rk 6 i 6 a + ks − k.
In each case, Had−s+1 (R)(gi,hi) 6= 0 if and only if
16i6

a + ks − k.
g

5. Rational singularities
Let R be a normal domain, essentially of finite type over a field of characteristic
zero, and consider a desingularization f : Z −→ Spec R, i.e., a proper birational
morphism with Z a nonsingular variety. One says R has rational singularities if
Ri f∗ OZ = 0 for each i > 1; this does not depend on the choice of the desingularization f . For N-graded rings, one has the following criterion due to Flenner [Fl]and Watanabe [Wa1].
Theorem 5.1. Let R be a normal N-graded ring which is finitely generated overa field R0 of characteristic zero. Then R has rational singularities if and only if it
is Cohen-Macaulay, a(R) < 0, and the localization Rp has rational singularities foreach p ∈ Spec R r {R+ }.
When R has an isolated singularity, the above theorem gives an effective criterion
for determining if R has rational singularities. However, a multigraded hypersurface
typically does not have an isolated singularity, and the following variation turns out
to be useful:
Theorem 5.2. Let R be a normal Nr -graded ring such that R0 is a local ring
essentially of finite type over a field of characteristic zero, and R is generated overR0 by elements
x11 , x12 , . . . , x1t1 ,x21 , x22 , . . . , x2t2 ,...,

xr1 , xr2 , . . . , xrtr ,where deg xij is a positive integer multiple of the i-th unit vector ei ∈ Nr . Then R
has rational singularities if and only if(1) R is Cohen-Macaulay,(2) Rp has rational singularities for each p belonging toSpec R \ V (x11 , x12 , . . . , x1t1 ) ∪ · · · ∪ V (xr1 , xr2 , . . . , xrtr ) ,and(3) a(R) < 0, i.e., a(Rϕi ) < 0 for each coordinate projection ϕi : Nr −→ N.
Before proceeding with the proof, we record some preliminary results.
Remark 5.3. Let R be an N-graded ring. We use R♮ to denote the Rees algebrawith respect to the filtration Fn = R>n , i.e.,R♮ = F0 ⊕ F1 T ⊕ F2 T 2 ⊕ · · · .

MULTIGRADED RINGS

13

When considering Proj R♮ , we use the N-grading on R♮ where [R♮ ]n = Fn T n . The
inclusion R = [R♮ ]0 ֒→ R♮ gives a map
f

Proj R♮ −→ Spec R .
Also, the inclusions Rn ֒→ Fn give rise to an injective homomorphism of gradedrings R ֒→ R♮ , which induces a surjectionπProj R♮ −→ Proj R .
Lemma 5.4. Let R be an N-graded ring which is finitely generated over R0 , andassume that R0 is essentially of finite type over a field of characteristic zero.
If Rp has rational singularities for all primes p ∈ Spec R r V (R+ ), then Proj R♮
has rational singularities.
Proof. Note that Proj R♮ is covered by affine open sets D+ (rT n ) for integers n > 1♮
and homogeneous elements r ∈ R>n . Consequently, it suffices to check that [RrTn ]0
has rational singularities. Next, note that
 ♮ 
1
1
RrT n 0 = R + [R]>n + 2 [R]>2n + · · · .
rr
In the case deg r > n, the ring above is simply Rr , which has rational singularities
by the hypothesis of the lemma. If deg r = n, then
 
 ♮ 
RrT n 0 = Rr >0 .
The Z-graded ring Rr has rational singularities and so, by [Wa1, Lemma 2.5], thering [Rr ]>0 has rational singularities as well.
Lemma 5.5. [Hy2, Lemma 2.3] Let R be an N-graded ring which is finitely generi
ated over a local ring (R0 , m). Suppose [Hm+R(R)] = 0 for all i > 0. Then, for+>0
all ideals a of R0 , one has ifor all i > 0 .
Ha+R+ (R) >0 = 0
We are now in a position to prove the following theorem, which is a variation of[Fl, Satz 3.1], [Wa1, Theorem 2.2], and [Hy1, Theorem 1.5].
Theorem 5.6. Let R be an N-graded normal ring which is finitely generated
over R0 , and assume that R0 is a local ring essentially of finite type over a field of
characteristic zero. Then R has rational singularities if and only if(1) R is Cohen-Macaulay,(2) Rp has rational singularities for all p ∈ Spec R r V (R+ ), and(3) a(R) < 0.
Proof. It is straightforward to see that conditions (1)–(3) hold when R has rational
singularities, and we focus on the converse. Consider the morphismf

Y = Proj R♮ −→ Spec R

14

KAZUHIKO KURANO, EI-ICHI SATO, ANURAG K. SINGH, AND KEI-ICHI WATANABE

as in Remark 5.3. Let g : Z −→ Y be a desingularization of Y ; the compositiongf

Z −→ Y −→ Spec R
is then a desingularization of Spec R. Note that Y = Proj R♮ has rational singularities by Lemma 5.4, sog∗ OZ = OYandRq g∗ OZ = 0

for all q > 1 .

Consequently the Leray spectral sequence
E2p,q = H p (Y, Rq g∗ OZ ) =⇒ H p+q (Z, OZ )degenerates, and we get H p (Z, OZ ) = H p (Y, OY ) for all p > 1. Since Spec R isaffine, we also have Rp (g ◦ f )∗ OZ = H p (Z, OZ ). To prove that R has rational
singularities, it now suffices to show that H p (Y, OY ) = 0 for all p > 1. Consider
the map π : Y −→ X = Proj R. We have
M
 p+1

.
H p (Y, OY ) = H p (X, π∗ OX ) =H p (X, OX (n)) = HR
(R)
+
>0

n>0

p
By condition (1), we have [Hm+R(R)]+p
implies that [HR
(R)]
+

>0

>0

= 0 for all p > 0, and so Lemma 5.5= 0 for all p > 0 as desired.

Proof of theorem 5.2. If R has rational singularities, it is easily seen that conditions (1)–(3) must hold. For the converse, we proceed by induction on r. The caser = 1 is Theorem 5.6 established above, so assume r > 2. It suffices to show that
RM has rational singularities where M is the homogeneous maximal ideal of R. Set
m = M ∩ R ϕr 0 ,and consider the N-graded ring S obtained by inverting the multiplicative set[Rϕr ]0 r m in Rϕr . Since RM is a localization of S, it suffices to show that S
has rational singularities. Note that a(S) = a(Rϕr ), which is a negative integerby (1). Using Theorem 5.6, it is therefore enough to show that RP has rational
singularities for all P ∈ Spec R r V (xr1 , xr2 , . . . , xrtr ). Fix such a prime P, and letψ : Zr −→ Zr−1
be the projection to the first r − 1 coordinates. Note that Rψ is the ring R regradedsuch that deg xrj = 0, and the degrees of xij for i < r are unchanged. Set 
p = P ∩ Rψ 0 ,and let T be the ring obtained by inverting the multiplicative set [Rψ ]0 r p in Rψ .
It suffices to show that T has rational singularities. Note that T is an Nr−1 -gradedring defined over a local ring (T0 , p), and that it has homogeneous maximal ideal
p + bT whereb = (Rψ )+ = (xij | i < r)R .

MULTIGRADED RINGS

15

Using the inductive hypothesis, it remains to verify that a(T ) < 0. By condition (1), for all integers 1 6 j 6 r − 1, we have
 ifor all i > 0 ,
HM (R)ϕj >0 = 0
and using Lemma 5.5 it follows that
 i
Hp+b (R)ϕj >0 = 0

for all i > 0 .

Consequently a(T ϕj ) < 0 for 1 6 j 6 r − 1, which completes the proof.



6. F-regularity
For the theory of tight closure, we refer to the papers [HH1, HH2] and [HH3].
We summarize results about F-rational and F-regular rings:
Theorem 6.1. The following hold for rings of prime characteristic.
(1) Regular rings are F-regular.
(2) Direct summands of F-regular rings are F-regular.
(3) F-rational rings are normal; an F-rational ring which is a homomorphic
image of a Cohen-Macaulay ring is Cohen-Macaulay.
(4) F-rational Gorenstein rings are F-regular.
(5) Let R be an N-graded ring which is finitely generated over a field R0 . If R
is weakly F-regular, then it is F-regular.
Proof. For (1) and (2) see [HH1, Theorem 4.6] and [HH1, Proposition 4.12] respectively; (3) is part of [HH2, Theorem 4.2], and for (4) see [HH2, Corollary 4.7],
Lastly, (5) is [LS, Corollary 4.4].
The characteristic zero aspects of tight closure are developed in [HH4]. Let K bea field of characteristic zero. A finitely generated K-algebra R = K[x1 , . . . , xm ]/ais of F-regular type if there exists a finitely generated Z-algebra A ⊆ K, and afinitely generated free A-algebra
RA = A[x1 , . . . , xm ]/aA ,such that R ∼= RA ⊗A K and, for all maximal ideals µ in a Zariski dense subsetof Spec A, the fiber rings RA ⊗A A/µ are F-regular rings of characteristic p > 0.
Similarly, R is of F-rational type if for a dense subset of µ, the fiber rings RA ⊗A A/µ
are F-rational. Combining results from [Ha, HW, MS, Sm] one has:
Theorem 6.2. Let R be a ring which is finitely generated over a field of characteristic zero. Then R has rational singularities if and only if it is of F-rational type.
If R is Q-Gorenstein, then it has log terminal singularities if and only if it is of
F-regular type.

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KAZUHIKO KURANO, EI-ICHI SATO, ANURAG K. SINGH, AND KEI-ICHI WATANABE

Proposition 6.3. Let K be a field of characteristic p > 0, and R an N-gradednormal ring which is finitely generated over R0 = K. Let ω denote the gradedcanonical module of R, and set d = dim R.
Suppose R is F-regular. Then, for each integer k, there exists q = pe such thatdrankK Rk 6 rankK [Hm(ω (q) )]k .

Proof. If d 6 1, then R is regular and the assertion is elementary. Assume d > 2.
ddLet ξ ∈ [Hm(ω)]0 be an element which generates the socle of Hm(ω). Since the mapω [q] −→ ω (q) is an isomorphism in codimension one, F e (ξ) may be viewed as andelement of Hm(ω (q) ) as in [Wa2].
Fix an integer k. For each e ∈ N, set Ve to be the kernel of the vector spacehomomorphism d (pe ) ,
where c 7−→ cF e (ξ) .
ω
(6.3.1)
Rk −→ Hmk
If cF e+1 (ξ) = 0, then F (cF e (ξ)) = cp F e+1 (ξ) = 0; since R is F-pure, it follows thatcF e (ξ) = 0. Consequently the vector spaces Ve form a descending sequence
V1 ⊇ V2 ⊇ V3 ⊇ · · · .
TThe hypothesis that R is F-regular implies e Ve = 0. Since each Ve has finite rank,
Ve = 0 for e ≫ 0. Hence the homomorphism (6.3.1) is injective for e ≫ 0.
We next record tight closure properties of general N-graded hypersurfaces. The
results for F-purity are essentially worked out in [HR].
Theorem 6.4. Let A = K[x1 , . . . , xm ] be a polynomial ring over a field K of
positive characteristic. Let d be a nonnegative integer, and set M = d+m−1− 1.
dM
Consider the affine space AK parameterizing the degree d forms in A in which xd1
occurs with coefficient 1.
Let U be the subset of AM
K corresponding to the forms f for which A/f A F-pure.
Then U is a Zariski open set, and it is nonempty if and only if d 6 m.
Let V be the set corresponding to forms f for which A/f A is F-regular. Then V
contains a nonempty Zariski open set if d < m, and is empty otherwise.
Proof. The set U is Zariski open by [HR, page 156] and it is empty if d > m by [HR,Proposition 5.18]. If d 6 m, the square-free monomial x1 · · · xd defines an F-pure
hypersurface A/(x1 · · · xd ). A linear change of variables yields the polynomial
f = x1 (x1 + x2 ) · · · (x1 + xd )in which xd1 occurs with coefficient 1. Hence U is nonempty for d 6 m.
If d > m, then A/f A has a-invariant d − m > 0 so A/f A is not F-regular.
Suppose d < m. Consider the set W ⊆ AM
K parameterizing the forms f for which
A/f A is F-pure and (A/f A)x1 is regular; W is a nonempty open subset of AMK.
Let f correspond to a point of W . The element x1 ∈ A/f A has a power whichMULTIGRADED RINGS

17

is a test element; since A/f A is F-pure, it follows that x1 is a test element. Notethat x2 , . . . , xm is a homogeneous system of parameters for A/f A and that xd−11generates the socle modulo (x2 , . . . , xm ). Hence the ring A/f A is F-regular if andonly if there exists a power q of the prime characteristic p such that(d−1)q+1

x1∈/ (xq2 , . . . , xqm , f )A .

The set of such f corresponds to an open subset of W ; it remains to verify that
this subset is nonempty. For this, consider
f = xd1 + x2 · · · xd+1 ,which corresponds to a point of W , and note that A/f A is F-regular since(d−1)p+1

x1∈/ (xp2 , . . . , xpm , f )A .



These ideas carry over to multi-graded hypersurfaces; we restrict below to thebigraded case. The set of forms in K[x1 , . . . , xm , y1 , . . . , yn ] of degree (d, e) inwhich xd1 y1e occurs with coefficient 1 is parametrized by the affine space AN
K where e+n−1
N = d+m−1−1.
deTheorem 6.5. Let B = K[x1 , . . . , xm , y1 , . . . , yn ] be a polynomial ring over a field
K of positive characteristic. Consider the N2 -grading on B with deg xi = (1, 0) anddeg yj = (0, 1). Let d, e be nonnegative integers, and consider the affine space ANKparameterizing forms of degree (d, e) in which xd1 y1e occurs with coefficient 1.
Let U be the subset of AN
K corresponding to forms f for which B/f B is F-pure.
Then U is a Zariski open set, and it is nonempty if and only if d 6 m and e 6 n.
Let V be the set corresponding to forms f for which B/f B is F-regular. Then V
contains a nonempty Zariski open set if d < m and e < n, and is empty otherwise.
Proof. The argument for F-purity is similar to the proof of Theorem 6.4; if d 6 m
and e 6 n, then the polynomial x1 · · · xd y1 · · · ye defines an F-pure hypersurface.
If B/f B is F-regular, then a(B/f B) < 0 implies d < m and e < n. Conversely,
if d < m and e < n, then there is a nonempty open set W corresponding toforms f for which the hypersurface B/f B is F-pure and (B/f B)x1 y1 is regular. In
this case, x1 y1 ∈ B/f B is a test element. The socle modulo the parameter ideal(x1 − y1 , x2 , . . . , xm , y2 , . . . , yn )B/f B is generated by x1d+e−1 , so B/f B is F-regular
if and only if there exists a power q = pe such that(d+e−1)q+1

x1∈/ (xq1 − y1q , xq2 , . . . , xqm , y2q , . . . , ynq , f )B .

The subset of W corresponding to such f is open; it remains to verify that it isnonempty. For this, use f = xd1 y1e + x2 · · · xd+1 y2 · · · ye+1 .
18

KAZUHIKO KURANO, EI-ICHI SATO, ANURAG K. SINGH, AND KEI-ICHI WATANABE

References[CHTV] A. Conca, J. Herzog, N. V. Trung, and G. Valla, Diagonal subalgebras of bigraded algebras
and embeddings of blow-ups of projective spaces, Amer. J. Math. 119 (1997), 859–901.
[CH]

S. D. Cutkosky and J. Herzog, Cohen-Macaulay coordinate rings of blowup schemes
Comment. Math. Helv. 72 (1997), 605–617.

[Da1]

V. I. Danilov, The group of ideal classes of a completed ring, Math. USSR Sbornik 6[Da2](1968), 493–500.
V. I. Danilov, Rings with a discrete group of divisor classes, Math. USSR Sbornik 12

[Fl]
[GG]

H. Flenner, Rationale quasihomogene Singularitäten, Arch. Math. 36 (1981), 35–44.
A. V. Geramita and A. Gimigliano, Generators for the defining ideal of certain rational[GGH]

surfaces, Duke Math. J. 62 (1991), 61–83.
A. V. Geramita, A. Gimigliano, and B. Harbourne, Projectively normal but superabun-(1970), 368–386.

dant embeddings of rational surfaces in projective space, J. Algebra 169 (1994), 791–804.
[GGP]

A. V. Geramita, A. Gimigliano, and Y. Pitteloud, Graded Betti numbers of some embedded rational n-folds, Math. Ann. 301 (1995), 363–380.

[GW1]

S. Goto and K.-i. Watanabe, On graded rings. I, J. Math. Soc. Japan 30 (1978), 179–213.

[GW2]

S. Goto and K.-i. Watanabe, On graded rings. II (Zn -graded rings), Tokyo J. Math. 1(1978), 237–261.

[Ha]

N. Hara, A characterisation of rational singularities in terms of injectivity of Frobenius
maps, Amer. J. Math. 120 (1998), 981–996.

[HW]

N. Hara and K.-i. Watanabe, F-regular and F-pure rings vs. log terminal and log canon-[HH1]

ical singularities, J. Algebraic Geom. 11 (2002), 363–392.
M. Hochster and C. Huneke, Tight closure, invariant theory, and the Briançon-Skoda
theorem, J. Amer. Math. Soc. 3 (1990), 31–116.

[HH2]

M. Hochster and C. Huneke, F-regularity, test elements, and smooth base change, Trans.
Amer. Math. Soc. 346 (1994), 1–62.

[HH3]

M. Hochster and C. Huneke, Tight closure of parameter ideals and splitting in modulefinite extensions, J. Algebraic Geom. 3 (1994), 599–670.

[HH4]

M. Hochster and C. Huneke, Tight closure in equal characteristic zero, in preparation.

[HHR]

M. Herrmann, E. Hyry, and J. Ribbe, On the Cohen-Macaulay and Gorenstein propertiesof multigraded Rees algebras, Manuscripta Math. 79 (1993), 343–377.

[HIO]

M. Herrmann, S. Ikeda, and U. Orbanz, Equimultiplicity and blowing up: an algebraic[HR]study, Springer-Verlag, Berlin-New York, 1988.
M. Hochster and J. Roberts, The purity of the Frobenius and local cohomology, Adv. in[Hy1]

Math. 21 (1976), 117–172.
E. Hyry, Blow-up rings and rational singularities, Manuscripta Math. 98 (1999), 377–
390.

[Hy2]

E. Hyry, The diagonal subring and the Cohen-Macaulay property of a multigraded ring,Trans. Amer. Math. Soc. 351 (1999), 2213–2232.

[Lv1]

O. Lavila-Vidal, On the Cohen-Macaulay property of diagonal subalgebras of the Rees

[Lv2]

algebra, Manuscripta Math. 95 (1998), 47–58.
O. Lavila-Vidal, On the diagonals of a Rees algebra, thesis, Universitat de Barcelona,

[LvZ]

1999.
O. Lavila-Vidal and S. Zarzuela, On the Gorenstein property of the diagonals of the Rees
algebra, Collect. Math. 49 (1998), 383–397.

[LS]

G. Lyubeznik and K. E. Smith, Strong and weak F-regularity are equivalent for gradedrings, Amer. J. Math. 121 (1999), 1279–1290.

MULTIGRADED RINGS

19

[MS]

V. B. Mehta and V. Srinivas, A characterization of rational singularities, Asian J. Math.

[RSS]

1 (1997), 249–271.
P. Roberts, A. K. Singh, and V. Srinivas, Annihilators of local cohomology in characteristic zero, Illinois J. Math. 51 (2007), 237–254.

[STV]

A. Simis, N. V. Trung, and G. Valla, The diagonal subalgebra of a blow-up algebra, J.
Pure Appl. Algebra 125 (1998), 305–328.

[Si]
[Sm]

A. K. Singh, Veronese subrings and tight closure, Pacific J. Math. 192 (2000), 399–413.
K. E. Smith, F-rational rings have rational singularities, Amer. J. Math. 119 (1997),[Tr]

N. V. Trung, Diagonal subalgebras and blow-ups of projective spaces, Vietnam J. Math.
28 (2000), 1–15.

[Wa1]

K.-i. Watanabe, Rational singularities with k ∗ -action, in: Commutative algebra (Trento,[Wa2]1981), Lecture Notes in Pure and Appl. Math. 84, Dekker, New York, (1983), 339–351.
K.-i. Watanabe, F-regular and F-pure normal graded rings, J. Pure Appl. Algebra 71

159–180.

(1991), 341–350.
Department of Mathematics, Meiji University, Higashimita 1-1-1, Tama-ku,
Kawasaki-shi 214-8571, Japan
E-mail address: kurano@math.meiji.ac.jp
Department of Mathematics, Kyushu University, Hakozaki 6-10-1, Higashi-ku,
Fukuoka-city 812-8581, Japan
E-mail address: esato@math.kyushu-u.ac.jp
Department of Mathematics, University of Utah, 155 South 1400 East,
Salt Lake City, UT 84112, USA
E-mail address: singh@math.utah.edu
Department of Mathematics, Nihon University, Sakura-Josui 3-25-40, Setagaya,
Tokyo 156-8550, Japan
E-mail address: watanabe@math.chs.nihon-u.ac.jp