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#A28 INTEGERS 11 (2011)

NORMALITY, PROJECTIVE NORMALITY AND EGZ THEOREM

S. S. Kannan

Chennai Mathematical Institute, Plot No-H1, SIPCOT IT Park, Padur Post, Tamilnadu, India

kannan@cmi.ac.in S. K. Pattanayak

Chennai Mathematical Institute, Plot No-H1, SIPCOT IT Park, Padur Post, Tamilnadu, India

santosh@cmi.ac.in

Received: 5/18/09, Revised: 1/6/11, Accepted: 2/20/11, Published: 4/16/11

Abstract

In this note, we prove that the projective normality of (P_{(V)/G,L), the celebrated}

theorem of Erd¨os-Ginzburg-Ziv and normality of an affine semigroup are all equiv-alent, whereV is a finite dimensional representation of a finite cyclic groupGover

C_{andLis the descent of the line bundleO(1)}⊗|G|_.

1. Introduction

LetV be a finite dimensional representation of a finite cyclic groupGover the field of complex numbersC_{. LetLdenote the descent of the line bundleO(1)}⊗|G|_{to the}

GIT quotientP_{(V)/G. In [4], it is shown that (}P_{(V)/G,L) is projectively normal.}

Proof of this uses the well known arithmetic result due to Erd¨os-Ginzburg-Ziv (see [2]).

In this note, we prove that the projective normality of (P_{(V)/G,L), the Erd¨}

os-Ginzburg-Ziv theorem and normality of an affine semigroup are all equivalent.

2. Preliminaries

Normality of a Semigroup: An affine semigroupM is a finitely generated sub-semigroup ofZn _{containing 0 for some n. LetN be the subgroup ofZ}n _generated

byM. Then,M is called normal if it satisfies the following condition: ifkx∈_M

INTEGERS 11 (2011) 2

For an affine semigroupMand a fieldKwe can form the affine semigroup algebra K[M] in the following way: as aK-vector space,K[M] has a basis consisting of the symbols Xa_{, a∈M, and the multiplication onK[M] is defined by theK-bilinear}

extension ofXa_.Xb_=Xa+b_.

We recall the following theorem from page 141, theorem 4.40 of [1].

Theorem 1. Let M be an affine semigroup, andK be a field. ThenM is normal if and only if K[M]is normal, i.e., it is integrally closed in its field of fractions.

Projective Normality: A polarized variety (X,L) where L is a very ample line bundle is said to be projectively normal if its homogeneous coordinate ring

⊕_n∈Z≥0H

0_(X,L⊗n_{) is integrally closed and is generated as aC-algebra byH}0_(X,L)

( see Exercise 5.14, Chapter II, Hartshorne [3]).

3. Main Theorem

In this section we will prove our main theorem.

Theorem 2. The following are equivalent

1. Erd¨os-Ginzburg-Ziv theorem: Let(a1, a2,· · ·, am), m≥2n−1 be a sequence of elements of Z_/nZ_{. Then there exists a subsequence (ai1, ai2,· · · , ai}

n) of length n

whose sum is zero.

2. LetGbe a cyclic group of ordernandV be any finite dimensional representation of G over C_{. Let L be the descent of O(1)}⊗n_{. Then (}P_{(V)/G,L) is projectively} normal.

2�_{. LetGbe a cyclic group of ordernandV be the regular representation ofGover} C_{. LetL be the descent ofO(1)}⊗n_{. Then(P(V)/G,L)is projectively normal.}

3. The sub-semigroup M of Zn _{generated by the set S = {(m}

0, m1,· · ·, mn−1) ∈(Z_≥0)n_:�n−1

i=0 mi=n and �n−1i=0 imi≡0mod n}is normal. Proof. We first prove (1), (2), and (2�_{) are equivalent.}

(1)⇒(2): This follows from the arguments given in page 2, paragraph 6 of [4]. (2)⇒(2�_{): This is straightforward.}

(2�_{)⇒(1): Let G=Z/nZ=< g >and let V be the regular representation ofG}

over C_{. Letξ be a primitive nth root of unity. Let{Xi :i= 0,1,· · ·, n}−1}be a basis ofV∗ _{given by:}

g.Xi =ξiXi, for everyi= 0,1,· · · , n−1.

By assumption the algebra⊕_d∈Z≥0(Sym

dn_V∗₎G _{is generated by (Sym}n_V∗₎G _(∗)

Let (a1, a2,· · ·, am), m≥2n−1 be a sequence of elements of G. Consider the

INTEGERS 11 (2011) 3

Take a = −(�2n−1_i=1 ai). Then (�2n−1i=1 Xai).Xa is a G-invariant monomial of

degree 2n, i.e., (�2n−1_i=1 Xai).Xa∈(Sym

2n_V∗₎G_.

By (∗), there exists a subsequence (ai1, ai2,· · · , ain) of (a1, a2,· · ·, a2n−1, a) of

lengthnsuch that�n_j=1Xa_ij isG-invariant. So,�nj=1aij = 0. Thus, we have the

implication.

We now prove (1) ⇒ _{(3) and (3)} ⇒ ₍₂�_{), which completes the proof of the}

theorem.

(1)⇒(3): LetNbe the subgroup ofZn_{generated byM. Suppose thatq(m}

0, m1, . . . ,

mn−1)∈M,q∈Nand (m0, m1,· · · , mn−1)∈N. We need to show that (m0, m1, . . . ,

mn−1)∈M.

Since q(m0, m1,· · ·, mn−1) ∈ M we have q.mi ≥ 0 ∀i. Hence, mi ≥ 0 ∀i.

Since N is the subgroup of Zn _{generated by M and M is the sub-semigroup of} Zn _{generated byS,N is generated byS as a subgroup ofZ}n_{. Therefore, the tuple}

(m0, m1,· · ·, mn−1) is an integral (not necessarily non-negative) linear combination

of elements ofS, i.e.,

(m0, m1,· · · , mn−1) = p

�

j=1

aj(m0,j, m1,j,· · ·, m(n−1),j),

whereaj ∈Zfor allj = 1,2,· · ·, pand (m0,j, m1,j,· · · , m(n−1),j)∈S. Therefore, n−1

�

i=0

mi = n−1 � i=0 p � j=1

ajmij = ( p

�

j=1

aj( n−1

�

i=0

mi,j)) = ( p

�

j=1

aj)n=kn

for some k∈Z_{. Moreoverk}≥0, sincemi≥0∀ i.

If k = 1 then�n−1_i=0 mi = n and hence, (m0, m1,· · · , mn−1) ∈ M. Otherwise

k≥2 and consider the sequence of integers

0, . . . ,0 � �� �

m0times

, 1, . . . ,1 � �� �

m1times

, · · ·, n−1, . . . , n−1

� �� �

mn−1times

This sequence has atleast 2nterms, since �n−1_i=0 mi =kn, k ≥2 and the sum of

it’s terms is divisible by n by the assumption that �n−1_i=0 imi ≡ 0mod n. So by

(1) there exists a subsequence of exactlynterms whose sum is a multiple ofn, i.e., there exists (m�

0, m�1,· · ·, m�n−1)∈Zn≥0 withm�i≤mi, ∀isuch that�n−1i=0 m�i=n

and�n−1_i=0 im�

iis a multiple ofn. So (m�0, m1�,· · ·, m�n−1)∈M. Then, by induction

(m0, m1,· · ·, mn−1)−(m�0, m�1,· · ·, m�n−1)∈M and, hence (m0, m1,· · · , mn−1)∈

M as required.

(3)⇒₍₂�_{): The polarized variety (P(V)/G,L) isP roj(⊕}

d∈Z≥0(H

0_(P(V),O(1)⊗d|G|₎G₎

which is the same as P roj(⊕_d∈Z≥0(Sym

d|G|_V∗₎G_{). Let R := ⊕}

d≥0Rd; Rd :=

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WriteV∗ _=⊕n−1

i=0CXi, where {Xi :i = 0,1,· · ·, n−1}is a basis of V∗ given by:

g.Xi =ξiXi, for everyi= 0,1,· · ·, n−1.

Let R� _{be the C-subalgebra of C[V] generated by R}

1 = (SymnV∗)G. We first

note that {Xm0

0 .Xm

1

1 . . . X mn−1

n−1 : (m0, m1,· · ·, mn−1) ∈ M} is a C-vector space

basis forR�_{. We now define the map}

Φ:C_[M]→R� _{by extending linearly the map}

Φ(X(m0,m1,···,mn−1)_{) =X}m0

0 .Xm

1

1 . . . X mn−1

n−1 for (m0, m1, . . . , mn−1)∈M.

ClearlyΦis a homomorphism ofC_{-algebras. Since{X}(m0,m1,...,mn−1)_{: (m}

0, m1, . . . ,

mn−1)∈M}is aC-vector space basis forC[M] and{X0m0.Xm

1

1 . . . X mn−1

n−1 : (m0, m1,

. . . , mn−1)∈M}is aC-vector space basis forR�,Φis an isomorphism ofC-algebras.

HenceR� _{is the semigroup algebra corresponding to the a}

ffine semigroupM. Since by assumption M is a normal affine semigroup, by Theorem 1 the algebra R� _is

normal. Thus, by Exercise 5.14(a) of [3], the implication (3)⇒(2�_{) follows.}

References

[1] W. Bruns, J.Gubeladze, Polytopes, Rings, and K-Theory. Springer Monographs in Mathemat-ics. Springer, Dordrecht, 2009.

[2] P. Erd¨os, A.Ginzburg, A.Ziv, A theorem in additive number theory, Bull. Res. Council, Israel, 10 F(1961) 41-43.

[3] R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.

[4] S. S. Kannan, S.K.Pattanayak, Pranab Sardar, Projective normality of finite groups quotients. Proc. Amer. Math. Soc. 137 (2009), no. 3, pp. 863-867.