seminormality and the cohen-macaulay propertypingli/thesis.pdf · in the early 1970’s , hochster...

Seminormality and theCohen-Macaulay Property

by

PING LI

A thesis submitted tothe Department of Mathematics and Statistics

in conformity with the requirementsfor the degree of Doctor of Philosophy

Queen’s UniversityKingston, Ontario, Canada

December 2004

Copyright c©Ping Li, 2004

i

Abstract

In the early 1970’s , Hochster proved that normal semigroup rings generated by

monomials are Cohen-Macaulay. When we weaken normal to seminormal, it is of

interest to ask the coincidence of seminormality and the Cohen-Macaulay property

of affine semigroup rings.

To solve this problem, three topics are discussed in this thesis.

(1) The Cohen-Macaulay property of certain affine semigroup rings;

(2) The seminormality of affine semigroup rings;

(3) The relation between the Cohen-Macaulay property and seminormality of

affine semigroup rings.

The main results of this thesis are:

(1) Let S ⊆ Nn be a simplicial affine semigroup. We give a practical criterion

to indicate the Cohen-Macaulay property of a simplicial affine semigroup

ring in terms of the cardinality of the spanning monomial set of the corre-

sponding semigroup. This is result is extremely useful because it provides

a significant algorithm which can be programmed by using Mathematica.

(2) We show that, under some hypotheses, the seminormality of an arbitrary

affine semigroup ring can be characterized by an extension S ′ introduced

by Goto, Suzuki and Watanabe.

(3) We prove that the Cohen-Macaulay property and seminormality coincide

under certain hypotheses. If the affine semigroup S contains all interior

lattice points of the group in the cone it spans, and the affine semigroup

ringR = k[S] is Cohen-Macaulay, thenR is seminormal. On the other hand,

if the rank of S is less than or equal to 3, S = S ′, and R is seminormal,

then R is Cohen-Macaulay.

ii

Acknowledgements

This thesis could not have been written without the help of many people. It is

my pleasure to thank and acknowledge all the significant people in my life for their

help during the graduate school years in Queen’s University.

My first and deepest gratitude must be to my supervisor Dr. Leslie Roberts.

I would like to thank him for his time, ideas and patient guidance throughout my

study and the writing of this thesis. The benefits of good supervision have extended

to my entire mathematical life, and have made my time in Queen’s University more

enjoyable.

I am also grateful for the help given to me by Dr. Tony Geramita and Dr. Les

Reid. Thanks also go to the members of my supervising committee.

I would like to thank all my friends in Canada for your support.

Many thanks to Queen’s University, in particular the Department of Mathe-

matics and Statistics, for their hospitality and for financially supporting my study

during the past four and half years. A special thanks goes to our secretary, Mrs.

Jennifer Read.

Finally, I dedicate this thesis to my parents, my husband and my son for their

endless love and support.

iii

Statement of Originality

All results presented in Chapter 3, 4, 5 of this thesis are original, unless otherwise

stated. The computer program TotalProgram in Appendix B is originally designed.

The results quoted from the literature are presented as statements with indicated

reference for their proof.

Contents

Abstract i

Acknowledgements ii

Statement of Originality iii

Notation vi

Chapter 1. Introduction 1

Chapter 2. Literature Review 10

2.1. Semigroups and Semigroup Rings 10

2.2. Multi-Graded Rings 18

2.3. Cohen-Macaulay Rings 21

2.4. Normal and Seminormal Semigroups and Semigroup Rings 26

Chapter 3. The Cohen-Macaulay Property of Semigroup Rings 29

3.1. The Cohen-Macaulay Property of Semigroup Rings 29

3.2. Socle and Type of a Special Class of Cohen-Macaulay Semigroup Rings 46

Chapter 4. Seminormality of Semigroups and Semigroup Rings 53

4.1. A New Criterion for Seminormal Semigroups 53

4.2. The Coincidence of CM and SN of Simplicial Semigroup Rings 56

4.3. Seminormality of Affine Semigroups 65

4.4. The Coincidence between SN and CM of Arbitrary Semigroup Rings 66

Chapter 5. The Canonical Module 73

iv

v

5.1. The Canonical Module 73

5.2. The Canonical Module of a Cohen-Macaulay Seminormal Semigroup

Ring 75

5.3. The Structure of KS 80

Chapter 6. Summary and Future Work 82

Appendix A. Some Criteria for the Cohen-Macaulay Property 84

A.1. Using the Definition of Cohen-Macaulay Rings 84

A.2. Using the Graded Auslander-Buchsbaum Theorem 87

A.3. By the Cardinality of the Spanning Monomial Set of a Certain

Semigroup Ring 89

Appendix B. Computer Program - TotalProgram 92

B.1. The Interior Lattice Points of a Cone Spanned by a Simplicial

Semigroup 93

B.2. The Hilbert Basis 99

B.3. The Spanning Set 102

B.4. The Cohen-Macaulay Property 105

Bibliography 108

Curriculum Vitae 112

vi

Notation

We always use N, Z, Q, and R to denote the sets of non-negative integers,

integers, rational numbers, and real numbers respectively.

• Notation for semigroups and semigroup rings:

S – a positive affine semigroup (contained in Zn)

C(S) – the convex rational polyhedral cone spanned by S in Qn.

G(S) – the additive group in Zn generated by S.

Hilb(S) – Hilbert basis of the semigroup S.

α1, · · · , αm – spanning vectors of the cone C(S) and α1, · · · , αm ∈ Hilb(S).

I = {1, 2, · · · ,m}

r = rankZG(S) defined to be the rank of the semigroup S, rank(S).

R = k[S] – the affine semigroup ring generated by S.

W – the subsemigroup of S generated by spanning vectors α1, · · · , αm of C(S).

B = k[W ] – the subsemigroup ring of R = k[S] generated by W .

• We shall adhere to the following notation.

A =

α1...αm

where α1, · · · , αm are spanning vectors of C(S).

Box = {α ∈ G(S) | α =∑m

i=1 λiαi, 0 6 λi 6 1, λi ∈ Q} if the semigroup S is

simplicial.

Ci = G(S) \ Si◦F – the relative interior of a face F of the cone C(S).

{Fi} – the codimension 1 faces (facets) of C(S), i ∈ I.

G(S)fi>0 = {α ∈ G(S) | fi(α) > 0} where fi is a linear function fi : Qn → Q.

vii

GJ – for a subset J of I, GJ = ∩i/∈JSi \ ∪i∈JSj.

Hi – a bounding hyperplane in Qn of the cone C(S) defined by the linear function

fi : Qn → Q, i.e., Hi = {α ∈ Qn | fi(α) = 0 }

H+i – the non-negative half space of the hyperplane Hi.

H−i – the non-positive half space of the hyperplane Hi.

KS = − ∩mi=1 Ci

Qn>0 = {(a1, · · · , an) ∈ Qn | ai > 0 for all i}.

Qn+ = {(a1, · · · , an) ∈ Qn | ai > 0 for all i}.

Rn>0 = {(a1, · · · , an) ∈ Rn | ai > 0 for all i}.

Rn+ = {(a1, · · · , an) ∈ Rn | ai > 0 for all i}.

S – the normalization of the semigroup S.

+S – the seminormalization of the semigroup S.

Si = {ω ∈ G(S) |ω + β ∈ S for some β ∈ Fi ∩ S}

S ′ = ∩mi=1Si

T = {t ∈ S | t cannot be written as the form t = ω + α, ω ∈ W \ {0}, α ∈ S}= {t ∈ S | t− ω /∈ S, for all ω ∈ W \ {0}}= {t ∈ S | t− αi /∈ S, i ∈ I}

Let U,U1, U2 be subsets of G(S).

U1 + U2 = {u1 + u2 |u1 ∈ U1 and u2 ∈ U2}

−U = {−u |u ∈ U}

XU – the vector subspace of k[S] whose k-basis consists the set of elements xu

with u ∈ U .

V – the Q-linear subspace generated by spanning vectors of the cone C(S).

Zr = Zn ∩ V . In fact, Zr ∼= Zr.

Zn>0 = {(a1, · · · , an) ∈ Zn | ai > 0 for all i}.

viii

πJ – the simplicial complex of non empty subsets P of J with the property

∩i∈P (S ∩ Fi) 6= (0).

CHAPTER 1

Introduction

The main problem that motivates this thesis is to study cohomological properties

of various affine semigroup rings and to relate these to topological, geometric and

combinatorial properties of the semigroups in the cone that they span.

Let S be an affine semigroup (a finitely generated semigroup containing 0 which

can be embedded in Zn for some positive integer n). We form the semigroup algebra

k[S] with basis S over a field k, and denote the k-basis element of k[S] which cor-

responds to α ∈ S by Xα. This notation is suggested by the fact that k[S] carries

a natural multiplication whose table is given by XαXα′ = Xα+α′ . The ring struc-

ture of k[S] can be easily verified under the above multiplication and the addition

k1Xα + k2X

α = (k1 + k2)Xα, k1, k2 ∈ k. For example, the affine semigroup algebra

k[Nn], corresponding to the affine semigroup Nn, is isomorphic to k[x1, · · · , xn], and

k[Zn] can be identified with Laurent polynomial ring k[x1, x−11 , · · · , xn, x−1

n ] upon

the choice of a basis for Zn. The rings k[S] where k is a field and S is an affine

semigroup are called affine semigroup rings. When an embedding S → Zn is given,

it induces an embedding k[S] → k[Zn], and the affine semigroup ring k[S] is merely

a subring of the Laurent polynomial ring k[x1, x−11 , · · · , xn, x−1

n ] generated by the

monomials Xα = xa11 · · ·xan

n , where α = (a1, · · · , an) ∈ S.

Connections between the semigroup S and its corresponding semigroup ring R

are strong. Hence studying the ring properties of affine semigroup rings in terms

of their corresponding semigroups formed by the exponent vectors of the monomial

1


basis of such rings would be very useful, and especially, the Cohen-Macaulay prop-

erty and seminormality deserve more attention. Therefore we may ask the following

questions.

Question:

(1) When is an affine semigroup ring Cohen-Macaulay? Furthermore, find cri-

teria for the Cohen-Macaulay property in terms of the corresponding semi-

group.

(2) When is an affine semigroup ring seminormal?

Note that a stronger version of Question 1 was posed by Bruns, Gubeladze and

Trung ([6] Problem 10).

The first step toward Question 1 is an important criterion given by Hochster in

the early 1970s. He [24] proved that normal semigroup rings generated by mono-

mials are Cohen-Macaulay by using the shellability of convex polytopes assembled

by the face lattice of the corresponding semigroups. (A domain is normal if it is

Noetherian and integrally closed in its fraction field). A purely ring-theoretic proof

was provided by Goto and Watanabe [18], in which a complex was constructed from

the corresponding semigroup for the local cohomology of the semigroup ring. Such

complexes, or their graded k-duals have been constructed by several authors, see

Trung and Hoa [46] and [39] by Schafer and Schenzel. (For normal and normality,

see chapter 2 section 3, or [27] by Matsumura for a detailed discussion.)

Now the first question can be refined as follows.

Question: When is a non-normal affine semigroup ring Cohen-Macaulay?

In attempting to solve this question, many algebraic criteria have been intro-

duced.

In 1976, Goto, Suzuki and Watanabe [16] characterized simplicial affine semi-

group rings by a suitable extension S ′ and claimed that a simplicial affine semigroup


ring R = k[S] is Cohen-Macaulay if and only if S = S ′. In a later paper given by

Goto and Watanabe [18], they extended this idea to more general affine semigroup

rings, namely standard semigroup rings. However, in 1986, a counter example to

the result of Goto and Watanabe was presented by Trung and Hoa in [46] using

Rees algebras. They pointed out that the condition S = S ′ is not sufficient for

the Cohen-Macaulay property of an affine semigroup ring k[S]. To get a correct

criterion, a topological condition on the convex cone spanned by its corresponding

semigroup should be added.

Another achievement that should be mentioned is a (perhaps not very practical)

computational criterion which was given by Rosales and Garcıa-Sanchez [36] in

1998 for simplicial affine semigroup rings, via the intersection of Apery sets of the

extremal rays of the cone spanned by the semigroup.

Regarding the second question, a bit of historical information may be necessary

to keep track of the study of seminormal affine semigroups and seminormal affine

semigroup rings.

The study of seminormality (initially named by Traverso [44]) goes back at least

forty years. Original work focused on Picard Groups, for details, see [44], [38], [14],

[42], [2], [3].

The definition of seminormality was introduced in the late 1960s. Let R be a

reduced commutative Noetherian ring with total ring of fractions K. The Picard

group of R is denoted by Pic(R). The following statements are equivalent.

(1) R ⊂ K is seminormal in K.

(2) The monomorphism Pic(R) → Pic(R[X]) induced by the inclusion R →R[X] is an isomorphism.

(3) If x ∈ K and x2, x3 ∈ R, then x ∈ R.

(4) If x ∈ K and xd ∈ R for all sufficiently large d ∈ Z>0, then x ∈ R.


The equivalence of (1) and (2) was proved by Traverso in [44] based on early

work of Endo [10], Bass and Murthy [5]. The equivalence of (1) and (3) was initially

showed by Hamann [22]. The equivalence of (1) and (4) can be obtained from

[44] Lemmas 1.3 and 1.7 and was given explicitly by Gilmer and Heitmann ([14]

Theorem 1), while Rush showed the equivalence of (3) and (4) in [38]. An overview

of seminormal rings was given by Swan in his paper “On Seminormality” [42].

In 1988, when Gubeladze worked on Anderson’s conjecture ([1], page 11) which

concerned the freeness of the finitely generated projective modules over a normal

semigroup algebra, he obtained a description of the maximal class of commutative

semigroups satisfying the cancellation condition for which all finitely generated pro-

jective modules over the corresponding semigroup algebra are free, and this class

turned to be the seminormal semigroups.

Definition 1.0.1: A semigroup S is said to be seminormal if for any α in the

group G a sufficient condition for α to belong to S is that 2α and 3α belong to S.

Gubeladze also pointed out the following result.

Theorem 1.0.2: ([1], Corollary 3.2) Let R be any principal ideal domain, and

S be any commutative monoid satisfying the cancellation condition and having no

nontrivial torsion. Then S is seminormal if and only if R[S] is seminormal.

We will use the notation in [44]. The seminormalization of an affine semigroup

S is defined as

+S = {α ∈ G(S) | dα ∈ S, d ∈ N, d >> 0},

where G(S) denotes the group generated by the affine semigroup S. Recall the

normalization of S is defined as

S = {α ∈ G(S) | dα ∈ S for some d ∈ Z>0}.

Compared with normal semigroups, seminormal semigroups enjoy many similar

properties. For instance, geometrically, both normal and seminormal semigroups


can be defined by finitely generated rational cones:

S = G(S) ∩ C(S).+S =

⋃F G(F ∩ S)∩

◦F, where F ranges over all faces of the cone C(S).

It is more convenient to use these geometrical descriptions to determine the normal-

ity and seminormality of affine semigroup rings. The first one is well known. The

second one can be found in [20], [32] or [43]. The result we are using is cited from

the paper [32] by Les Reid and Leslie Roberts.

Geometrically, seminormal and normal semigroups are so close that it is of in-

terest to consider whether the ring properties of normal semigroup rings can be

extended to seminormal semigroup rings. We know that a normal affine semigroup

ring is always Cohen-Macaulay. A question is raised naturally.

Question: When is an seminormal affine semigroup ring Cohen-Macaulay?

Conversely, we could also ask when a Cohen-Macaulay affine semigroup ring is semi-

normal?

The following example is evidence that the Cohen-Macaulay property and semi-

normality may sometimes be coincident.

Example 1.0.3: A = k[x2, xy, y] is found to be both Cohen-Macaulay and

seminormal. �

Some work on projective curves has been done regarding to the coincidence

between Cohen-Macaulay and seminormal. For instance, A. V. Geramita and C. A.

Weibel [13] (Corollary 5.9) showed the following theorem.

Theorem 1.0.4: Let X be a reduced closed subscheme of Prk = ProjR (k a field

and R = k[x0, · · · , xr], r > 2). Let A be the homogeneous co-ordinate ring of X. If

X is a connected union of lines which have linearly independent directions at each

intersection point then A is seminormal if and only if A is Cohen-Macaulay. �


A further development of this idea was a criterion given by Leslie G. Roberts

and Balwant Singh ([33] Corollary 3.5).

Theorem 1.0.5: Let X be a reduced closed subscheme of Prk, and A its ho-

mogeneous co-ordinate ring. Assume that k is algebraically closed and that X is

connected and seminormal as a scheme, dimX = 1. Then A is seminormal if and

only if A is Cohen-Macaulay. �

However, the relationship between the Cohen-Macaulay property and seminor-

mality of affine semigroup rings is less well known. Moreover, the task is not eas-

ily accomplished. For affine semigroup rings lots of examples show that Cohen-

Macaulay and seminormal may not coincide. For example, A = k[t2, t3] is Cohen-

Macaulay but not seminormal, conversely, A = k[x, y, z2, xz, yz] is seminormal but

not Cohen-Macaulay.

According to the geometric description of seminormal affine semigroups, if S

is seminormal then S contains all points of G(S)∩◦

C(S), where G(S) is the group

generated by S in Zn and◦

C(S) is the relative interior of the cone C(S) spanned by S

in Qn. Hence studying the Cohen-Macaulay property of semigroup rings associated

with such semigroups may give us a key for finding the relations between Cohen-

Macaulay and seminormal. In this thesis, we put our emphasis on such concrete

affine semigroup rings.

This thesis is divided into five chapters and two appendixes. In chapter 1, we

present the motivations, history of the questions, and summarize the main results

of my research.

In chapter 2, we present the mathematical foundations for this thesis. one can

find basic theory of semigroup and semigroup rings, as well as some results on ring

properties.

In chapter 3, a practical method is introduced to characterize the Cohen-Macaulay

property of simplicial affine semigroup rings.


Theorem 1.0.6: (Theorem 3.1.12) Let S ⊆ Nn ∩ C be a simplicial affine

semigroup. Let spanning vectors of the cone C(S) be α1, · · · , αm. rank(S) = m 6

n. Let the index of the group G(S) in Zm be h. Then R is Cohen-Macaulay if

and only if the cardinality of the spanning monomial set XT is det(A)/h where

A =

α1...αm

. �

We shall show how this method is specially convenient when we explore the

coincidence between the Cohen-Macaulay property and seminormality of seminormal

simplicial affine semigroup rings in chapter 4.

Then, in section 2 of chapter 3, we deduce our application on the case of simplicial

standard affine semigroup rings containing all integer lattice points of Zn>0 and give

a formula to calculate the type of such rings.

Our focus is put on Chapter 4 in which the coincidence between the Cohen-

Macaulay property and seminormality is presented. First we start with the sim-

plicial case. Using above theorem and a new criterion introduced in section 1 for

seminormality of simplicial affine semigroup rings, one direction can be obtained

without difficulty.

Theorem 1.0.7: (Theorem 4.2.1) Let S ⊆ Nn be a simplicial affine semigroup

with G(S)∩◦

C(S)⊂ S. Then R = k[S] Cohen-Macaulay implies R seminormal. �

For the other direction, applying Goto, Suzuki and Watanabe’s method, i.e., a

simplicial affine semigroup ring k[S] is Cohen-Macaulay if and only if S = S ′, where

S ′ is a suitable extension of the semigroup S, we have the following result.

Theorem 1.0.8: (Corollary 4.2.15) Let S ⊆ Nn be a seminormal simpli-

cial affine semigroup. Then R = k[S] is Cohen-Macaulay if and only if S∩◦F=⋂

j∈J G(S∩Fj)∩◦F for every proper face F = ∩i∈JFi of the cone C(S), where J ⊂ I

is the set of indexes of all facets containing the face F . �


In particular, when rank(S) 6 2, we have a desired result.

Theorem 1.0.9: (Theorem 4.2.12) Let S be a simplicial affine semigroup con-

taining all elements of G(S)∩◦

C(S) with rank(S) 6 2. Then R is Cohen-Macaulay

if and only if R is seminormal. �

Inspired by above phenomenon, we try to generalize these results to non-simplicial

semigroup rings. We find that for an arbitrary affine semigroup ring k[S] with

G(S)∩◦

C(S)⊂ S, S = S ′ implies seminormal. According to the theorem given by

Trung and Hoa ([46], Corollary 2.2), the following theorem holds.

Theorem 1.0.10: (Corollary 4.4.4) Let S ⊆ Nn be an arbitrary affine semi-

group with G(S)∩◦

C(S)⊂ S. If the semigroup ring R = k[S] is Cohen-Macaulay,

then R is seminormal. �

A conjecture is given in section 4,

Conjecture: Let S be an arbitrary affine semigroup. Then S seminormal and

S = S ′ imply the semigroup ring R = k[S] is Cohen-Macaulay. �

and it is solved when rank(S) 6 3.

Theorem 1.0.11: Let S be an arbitrary affine semigroup with rank r = 3.

Then S seminormal and S = S ′ imply the semigroup ring R = k[S] is Cohen-

Macaulay. �

In the last chapter, we will summarize the main results of my research. We will

also give a series of topics related to this thesis which provide ample motivation for

future work.

Appendix A introduces some useful criteria to determine whither or not a affine

semigroup ring is Cohen-Macaulay by using a computer. All examples in this thesis

are tested in Appendix A.


Many of the results in this thesis were inspired by a variety of examples gen-

erated by several computer programs including Macaulay 2 [19] by D. Grayson

and M. Stillman, Normaliz [8] by W. Bruns and R. Koch, Mathematica [47], and

TotalProgram - a computer program for generating seminormal affine semigroup

rings and calculating the Cohen-Macaulay property [26] by Ping Li. In Appen-

dix B the code of TotalProgram is provided. We also explain the mathematical

background of this program and demonstrate this program by some examples.

CHAPTER 2

Literature Review

In this chapter, we will construct the mathematical framework of affine semi-

groups and affine semigroup rings for later chapters. As a consequence, most of the

material in this chapter is well known.

The chapter is organized in four sections. The first section is devoted to a review

of affine semigroups and affine semigroup rings. In Section 2, we review multi-graded

rings and extend our discussion to positive affine semigroup rings. We shall see that

a positive affine semigroup ring can be considered as an Nn-graded ring and an

N-graded ring. Section 3 treats the Cohen-Macaulay property of affine semigroups

and affine semigroup rings. It also contains some definitions and basic results on

Canonical modules, and Buchsbaum rings. Various topics about normality and

seminormality will be presented in Section 4.

2.1. Semigroups and Semigroup Rings

This section will present basic terminology and notation relating to affine semi-

groups and affine semigroup rings.

2.1.1. Affine Semigroups. A semigroup is a nonempty set closed under an

associative binary operation. Let (S,+) be a semigroup, then S is commutative if it

is commutative under the operation +, and S has a neutral element (usually denoted

by 0) if there exists an identity with respect to +. A semigroup with identity is called

a monoid (note that all semigroups in this thesis are monoids. Therefore we shall

substitute semigroup for monoid). A semigroup S is finitely generated if there exists

10


a finite number of elements α1, · · · , αt ∈ S such that for any α ∈ S, α can be written

as a linear combination of these elements. An element α in a semigroup S is called

invertible if −α ∈ S.

Definition 2.1.1: A semigroup is called cancellative if α + β = α + γ implies

β = γ for α, β, γ ∈ S. �

Definition 2.1.2: An affine semigroup is a semigroup (always containing a

neutral element) which is finitely generated and can be embedded in Zn for some

n ∈ N. We call an affine semigroup S positive if 0 is the only invertible element in

S. �

Remark: Through out this thesis, all semigroups are commutative and can-

cellative positive affine semigroups with a neutral element.

A non-zero element α in a semigroup S is called irreducible if it cannot be

written as the sum of two other non-zero elements of S. Let S be an positive affine

semigroup. The embedding S → Zn implies that every element of S can be written

as the sum of the uniquely determined irreducible elements. Since S is finitely

generated, the set of irreducible elements is also finite. Then we call the minimal

system of irreducible elements of S the Hilbert basis, denoted by Hilb(S).

Let G(S) denote the group generated by an affine semigroup S. Then G(S) ∼= Zr

with r ∈ N which we call the rank of S, denoted by rank(S).

A subset K of Rd is convex if for any two points x, y ∈ K, the straight line

segment λx+ (1− λ)y, 0 6 λ 6 1, belongs to K.

Definition 2.1.3: A non-empty convex subset C of Rn is called a cone if it is

closed under non-negative real linear combinations. That is, for α1, · · · , αd ∈ C and

λi ∈ R+, we have∑d

i=1 λiαi ∈ C. �


A cone is finitely generated if it is generated by a finite set. A cone is r-

dimensional if the smallest linear subspace containing it is of dimension r.

In n-dimensional real vector space Rn, we assume the ordinary scalar product

ξη = a1b1 + · · ·+ anbn, for ξ = (a1, · · · , an), η = (b1, · · · , bn).

For a fixed α ∈ Rn, α 6= 0, define a linear function fα(ξ) : Rn → R by fα(ξ) = ξα

for ξ ∈ Rn. Given a fixed d ∈ R, the set H := {ξ | fα(ξ) = d} is a hyperplane.

H+ := {ξ | fα(ξ) > d} and H− := {x | fα(ξ) 6 d} are called the closed half-spaces

bounded by H. If d = 0, then H is a hyperplane through the origin.

Theorem 2.1.4: (Ziegler [48], Theorem 1.3, Lecture 1) A cone C ⊆ Rn is

finitely generated if and only if it is a finite intersection of closed linear half spaces.

�

Let C be a finitely generated cone. Then C = ∩mi=1H+i , where the H+

i are half

spaces. This family {H+i | i = 1, · · · ,m} is called irredundant provided ∩j 6=iH+

j 6= C

for each j = 1, · · · ,m. If {H+i | i = 1, · · · ,m} is irredundant, then Fi = C ∩Hi is a

facet of C. For an r-dimensional cone, the proper faces of dimensions 1 and r − 1

are called rays and facets. Clearly {0} is the only vertex in the cone.

Definition 2.1.5: For a semigroup S, the smallest cone containing S,

R+S = {n∑i=0

λiαi |λi ∈ R+, αi ∈ S}

is called the cone generated by S, denoted by C(S). �

We need some results related to the cone generated by an affine semigroup.

1. If the semigroup S is finitely generated, then the cone C(S) is finitely gener-

ated.

2. If rank of S is r, then the cone C(S) is r-dimensional.


Definition 2.1.6: A minimal subset of the Hilbert basis of an affine semigroup

S which spans the cone C(S) is called a set of spanning vectors of C(S). �

The proposition below is well known. A proof is presented here because this

proposition is essential to the thesis, and it is not obvious at all.

Proposition 2.1.7: Let S ⊂ Zn be an affine semigroup. Let spanning vectors

of C(S) be α1, · · · , αd ∈ S. Then every element in S can be written as a non-negative

rational linear combination of these spanning vectors.

Proof: Clearly the cone C(S) is spanned by α1, · · · , αd ∈ S over R. For every

α ∈ S ⊂ C(S), Caratheodory’s Theorem ( Ziegler [48], p46) guarantees that with

proper triangulation, α can be represented as a linear combination of a linearly

independent subset of α1, · · · , αd. That is

α =∑i

λiαi, λi ∈ R+.

Since the above linear system of equations in λi’s has integer coefficients, λi are

non-negative rational numbers. �

3. Define a rational cone Q+S to be the cone generated by S over Q, that is,

Q+S = {n∑i=0

λiαi |λi ∈ Q+, αi ∈ S}.

Obviously S = S ∩ R+S = S ∩Q+S, and G(S) ∩ R+S = G(S) ∩Q+S where G(S)

is the group generated by S in Zn.

From now on, we shall consider the cone C(S) as an rational cone based on the

Proposition 2.1.7.

4. A subset C of Rn is a finitely generated rational cone if and only if it is the

intersection of finitely many rational half spaces.


2.1.2. Some Interesting Semigroups. In this thesis we usually talk about

the following two kinds of affine semigroups: simplicial and standard affine semi-

groups.

Definition 2.1.8: An affine semigroup S ⊆ Nn is called simplicial if the cone

C(S), which is generated by S in Rn, is spanned by m linearly independent vectors

α1, · · · , αm of S, where m = rankS. �

Geometrically, a simplicial semigroup means that its cone C(S) has m extremal

rays or equivalently, m facets.

Example 2.1.9: The following S in graphs (1) and (2) are simplicial, but not

the one in graph (3)

-

6

��

��

S

C(S)

Graph-1

-

6

��+

S

C(S)

Graph-2

-

6

��+

��

��

S

C(S)

Graph-3

Obviously, if S ⊆ Nn is a simplicial affine semigroup, the rank of S, rank(S) =

r 6 n.


Let S ⊆ Nn be an arbitrary affine semigroup. Recall the normalization of a

semigroup is defined as follows.

S = {α ∈ G(S) | lα ∈ S, for some l ∈ N \ {0} }.

Take

S(i) = {α = (a1, · · · , an) ∈ G(S) | ai = 0 }, i = 1, · · · , n.

Definition 2.1.10: An affine semigroup S ⊆ Nn is called standard if the

following conditions are satisfied:

(1) S = G(S) ∩ Nn.

(2) S(i) 6= S(j) for i 6= j.

(3) rank(G(S(i)) = r − 1, i = 1, · · · , n. �

Geometrically, these conditions tell us that the cone C(S) has exactly n (r− 1)-

dimensional facets lying on the coordinate planes xi = 0, for i = 1, · · · , n. Hence

S(i) = S ∩ Fi, where Fi is the i-th facet of the cone C(S).

Remark: In fact, every affine semigroup can be transformed isomorphically

onto a standard one according to Trung and Hoa. In their paper [46], they pointed

out this desired result by using Hochster’s Transformation [24].

Hochster’s Transformation:

Let S ⊂ Zn be a positive affine semigroup. rank(S) = r. By the discussion

before, the cone C(S) generated by S is the intersection of m rational half spaces

H+i , i = 1, · · · ,m, i.e., C(S) =

⋂mi=1H

+i . Then one can find m linear functions

fi : Qn → Q, i = 1, · · · ,m, corresponding with the hyperplanes H1, · · ·Hm such

that

C(S) = {α ∈ Qn | fi(x) > 0, i ∈ I}.

Then Fi = Hi

⋂C(S), i = 1, · · · ,m, are the (r − 1)-dimensional facets of C(S).


Let T denote the linear transformation T : Qn → Qm which sends every element

α ∈ Qn to the element T (α) = (f1(α), · · · , fm(α)) ∈ Qm. Clearly T (S) ⊂ Qm+ , in

other words, T sends the semigroup S to the first orthant of Qm, moreover, we may

assume T (S) ⊆ Nm by substituting a positive integer multiple of fi for fi.

Hochster [24] proved that T (S) is a positive semigroup isomorphic to S and

T (S) = G(T (S)) ∩ Nm.

According to Trung and Hoa, T also induces an isomorphism between the

semigroup S ∩ Fi and T (S)(i), i = 1, · · · ,m. Then S ∩ Fi 6= S ∩ Fj implies

T (S)(i) 6= T (S)(j) for i 6= j. And since rankG(T (S)(i)) = rankG(S ∩ Fi) = r − 1,

we can conclude that T (S) is a standard affine semigroup in Nn.

Notice that after the transformation, the index of the group G(T (S)) generated

by the semigroup T (S) may be changed. This can be seen by the following example.

Example 2.1.11: Let S ∈ Z2 be an affine semigroup with Hilbert basis

Hilb(S) = {(2, 1), (1, 1), (1, 2)}. It is easy to show that the index of G(S) in Z2

is 1.

Now take f1 = (2,−1), f2 = (−1, 2). Define a map T : Z2 → Z2 which is given

by T (α) = (f1α, f2α) for any α ∈ Z2. Applying this transformation to S, the

Hilbert basis of T (S) contains the elements T (2, 1) = (3, 0), T (1, 2) = (0, 3), and

T (1, 1) = (1, 1). Clearly T (S) is standard, and the index of group G(T (S)) in Z2 is

3. �

The following transformation has two consequences that make it extremely use-

ful in this thesis. First, it can convert every positive affine semigroup isomorphically

to the first orthant of Qn. Second, it will keep the index of the group after the

transformation. Note that the new semigroup may not be standard after the trans-

formation.

Unimodular Transformation


Let S ∈ Zn be an positive affine semigroup with rank r. Suppose that the

spanning vectors of the cone C(S) are α1, · · · , αm. Then in Zn, there must be

a unimodular vector v in Zn such that the cone stays in the positive side of the

hyperplane defined by v. This implies that for any element α ∈ S ⊂ C(S), the

scalar product αv is positive.

Assume a basis of Zn extending the vector v consists of vectors v1 = v, v2, · · · , vn.Take v1, v2, · · · , vn as the columns of the matrix M . Then M represents a morphism

φ : Zn → Zn given by φ(α) = αM for all α ∈ Zn. The map φ ensures that φ(S) is

also an affine semigroup with spanning vectors φ(α1), · · · , φ(αm), and that for any

α ∈ S, the first coordinate of the vector φ(α) ∈ φ(S) is positive.

Now chose suitable integers λ2, · · · , λn and build the matrix M∗ as follows.

M∗ =

1 λ2 λ3 · · · λn0 1 0 · · · 00 0 1 · · · 0

· · ·0 0 0 · · · 1

n×n

, λi ∈ Z>0, i = 2, · · · , n

such that for all spanning vectors of the semigroup φ(S), φ(αi)M∗ ∈ Nn.

The Matrix M∗ represents a map ψ : Zn → Zn as well. Consider the composition

of maps φ and ψ,

ψφ : Zn M // Zn M∗// Zn

Easy to show that φψ(S) is an affine semigroup which is isomorphic to S since both

φ and ψ are linear and convertible. Clearly G(ψφ(S)) ∼= G(S) and the index of the

group remains the same.

Now we may assume that all semigroups in this thesis are in the first orthant by

unimodular transformation.

2.1.3. Affine Semigroup Rings. Let k be a field. The rings R = k[S] with

S an affine semigroup are called affine semigroup rings. Let us recall the definition

of Krull-dimension.


Definition 2.1.12: Let R be an ring. Let M 6= 0 be an R-module. The Krull-

dimension of M , denoted by dimR(M), is the supremum of lengths of chains of

prime ideals in the support of M if this supremum exists. If M is finitely generated,

then dimR(M) = dimR[R/(0 : M)] �

The following facts hold.

(1) The semigroup ring k[S] has monomial k-basis Xα with α ∈ S.

(2) If S is finitely generated, then k[S] is finitely generated subalgebra of

k[x±1 , · · · , x±n ] generated by finitely many monomials. If S is positive, then

XHilb(S) is a minimal set of k-algebra generators for k[S].

(3) dimk(k[S]) = rank(S), since obviously rank(S) is the transcendence degree

of the fraction field of k[S].

Theorem 2.1.13: (R. Gilmer [15], Theorem 7.7) Let R be a unitary ring and

let S be a semigroup. the following conditions are equivalent.

(1) The semigroup ring R[S] is Noetherian.

(2) R is Noetherian and S is finitely generated.

�

2.2. Multi-Graded Rings

A theory of Zn-graded rings was developed by Goto and Watanabe [18], Stanley

[40]. Here we summarize the basic terminologies.

A Zn-graded ring is a ring R together with a decomposition R =⊕

s∈Zn Rs (as

a Zn-module) such that RsRt ⊂ Rs+t for all s, t ∈ Zn.

A module over a Zn-graded ring R is called a Zn-graded R-module if there is

a decomposition M =⊕

s∈Zn Ms (as a Zn-module) such that RsMt ⊂ Ms+t for all

s, t ∈ Zn. The elements x ∈ Ms are called homogeneous of degree s. An arbitrary


element x ∈ M has a unique presentation x =∑

s xs as a sum of homogeneous

elements xs ∈Ms.

An ideal of a Zn-graded ring R is homogeneous if it is generated by homogeneous

elements of R.

We say that a graded ring R is a positively multi-graded ring if R =⊕

s∈Nn Rs,

that is, R is Nn-graded. For example, R = k[x, y] is N2-graded.

If R is an Nn-graded ring for which R0 is a local ring with maximal ideal m0,

then m = m0 ⊕ (⊕s 6=0Rs) is the unique homogeneous maximal ideal of R. It follows

R/m = R0/m0, which is a field. Such a positively Nn-graded ring is denoted as

(R,m).

Remark:

1: Let S ⊆ Nn be a positive affine semigroup. Let R = k[S] be an affine

semigroup ring over a field k, R0 = k. Then R has a natural Nn-grading

R = ⊕α∈SRα: for α = (a1, · · · , an) ∈ S ⊆ Nn, we let Rα = {dXα =

dxa11 · · ·xan

n | d ∈ k} be the α-th homogeneous component of R.

Note that the unique maximal homogeneous ideal m of R is the ideal

generated by the monomials Xα, α ∈ S \ {0}, and R/m = k. �

2: By a unimodular transformation, every positive affine semigroup can be

transformed isomorphically into the first orthant. So we may say every

positive affine semigroup ring is isomorphic to a positively multi-graded

ring in above way.

With no loss of generality, let us assume that the semigroup S is in the first or-

thant. In order to use the results on Z-graded rings and modules for affine semigroup

rings we say that the affine semigroup ring R = k[S] has an admissible grading if

and only if S is positive. This can be understood as follows: if the semigroup S is


positive, then R has a decomposition

R =⊕i∈N

Ri

where each component Ri is a direct sum of finitely many Zn-graded components,

i.e.,

Ri = ⊕|α|=iRα, |α| = |(a1, · · · , an)| =n∑i=1

ai, for α ∈ Zn

The maximal homogeneous ideal m of R is the ideal generated by monomials Xα

with α 6= 0, α ∈ S.

Example 2.2.1: Let S ∈ N2 be an affine semigroup with Hilbert basis

Hilb(S) = {(1, 0), (1, 1), (0, 2), (0, 3)}.

The corresponding semigroup ring is R = k[x, xy, y2, y3] with deg(x) = (1, 0) and

deg(y) = (0, 1). Then R is N2-graded, i.e., R =⊕

(i,j)∈N2 R(i,j). Take an element

f = x3y ∈ R, clearly f ∈ R(3,1), therefore f is a homogeneous element of R with

degree (3, 1). The element g = x2y+xy2 is not N2-homogeneous because x2y ∈ R(2,1)

and xy2 ∈ R(1,2). However, g is a homogeneous element if we consider R as N-

graded. �

Definition 2.2.2: The homogeneous Krull-dimension of the graded ring R,

denoted dim(R), is the length of the longest chain of homogeneous prime ideals in

R. �

Definition 2.2.3: Let R be a graded ring. Let M 6= 0 be a graded R-module.

The graded Krull-dimension of M , denoted by dimR(M), is the supremum of lengths

of chains of homogeneous prime ideals in the support of M if this supremum exists.

If M is finitely generated, then dimR(M) = dim[R/(0 : M)] �

Definition 2.2.4: Let (R,m) be a positively graded ring, n = dim(R). A set

{f1, · · · , fn} of homogeneous elements of positive degree is called a homogeneous

system of parameters if the ideal (f1, · · · , fn) is m-primary. �


It is more convenient to use the theorem below as a definition of homogeneous

system of parameters.

Theorem 2.2.5: A system of elements {f1, · · · , fn} of R is a homogeneous

system of parameters if and only if R is finitely generated as a module over the

subalgebra A = k[f1, · · · , fn] generated by the fi, i = 1, · · · , n. �

2.3. Cohen-Macaulay Rings

In this section, we shall review definitions and some well-known results about

Cohen-Macaulay rings, Buchsbaum rings, and the canonical module of graded rings.

We will relate these results to semigroup rings. The material in this section can be

found in many books. I followed mostly “Cohen-Macaulay Rings” by W. Bruns and

J. Herzog [7].

2.3.1. Cohen-Macaulay Rings and Modules. Let M be a module over a

ring R. We say that f ∈ R is an M-regular element if fg = 0 for g ∈ M implies

g = 0, in the other words, if f is not a zero-divisor on M .

Definition 2.3.1: A sequence f = f1, · · · , fr of elements of the ring R is

called an M-regular sequence or simply an M -sequence if the following conditions

are satisfied:

(1) fi is an M/(f1, · · · , fi−1)M -regular element for i = 1, · · · , r;(2) M/fM 6= 0. �

Theorem 2.3.2: (Rees [28]) Let R be a Noetherian ring, M a finitely generated

R-module, and I an ideal such that IM 6= M . Then all maximal M -sequences in I

have the same length n given by n = min{i : ExtiR(R/I,M) 6= 0} �

This allows us to introduce the fundamental notion of depth.


Definition 2.3.3: Let M be a finitely generated module over a Noetherian ring

R, and I ⊆ R an ideal with IM 6= M . Then the depth of I on M is the maximal

length of an M -regular sequence in I, denoted by depth(I,M).

If R is a local (or positively graded) ring with a unique maximal (homogeneous)

ideal m, we write depth(M) for depth(m,M). �

Definition 2.3.4: Let R be a Noetherian local ring. A finitely generated R-

module M 6= 0 is a Cohen-Macaulay module if depth(M) = dim(M). If R itself

a Cohen-Macaulay module, then it is called a Cohen-Macaulay ring. A maximal

Cohen-Macaulay module is a Cohen-Macaulay module M such that dim(M) =

dim(R).

In general, if R is an arbitrary Noetherian ring, then M is a Cohen-Macaulay

module if Mm is a Cohen-Macaulay module for all maximal ideals m ∈Supp(M).

For M to be a maximal Cohen-Macaulay module if Mm is such an Rm-module for

each maximal ideal m of R. As in the local case, R is a Cohen-Macaulay ring if it

is a Cohen-Macaulay module. �

The following is an alternative definition of Cohen-Macaulay rings.

Definition 2.3.5: A local ring R is called a Cohen-Macaulay ring if some

(equivalently, every) system of parameters of R is a regular sequence. �

A graded version of this definition can be found in the following theorem.

Theorem 2.3.6: (Derkson and Kemper [9], Proposition 2.5.3) Let R be a

Noetherian N-graded algebra over a field k with k = R0 the homogeneous part of

degree 0. Then the following conditions are equivalent:

(1) R is Cohen-Macaulay;

(2) Every homogeneous system of parameters is R-regular;


(3) If f1, · · · , fn is a homogeneous system of parameters, then R is a free module

over k[f1, · · · , fn];(4) There exists a homogeneous system of parameters f1, · · · , fn such that R

is a free module over k[f1, · · · , fn]. �

Remark: Let (R,m) be a positively graded ring, then R is a Cohen-Macaulay

ring if and only if Rm is Cohen-Macaulay for the homogeneous maximal ideal m.

Then the equivalence between (1) and (4) is equivalent to Definition 2.3.5.

Let R be a ring, and M an R-module. M has an augmented projective resolution

P. : · · · // Pnφn // Pn−1

// · · · // P1

φ1 // P0

φ0 // M // 0.

Set M0 = M and Mi = kerφi−1 for i > 1. Note that M determines Mi up to

isomorphism as graded R-modules. Mi is called the i-th syzygy of M . The projective

dimension of M , denoted by projdim(M), is the least integer n for which Mn is

projective; replacing Pn by Mn one gets a projective resolution of M of length n:

0 // Mn// Pn−1

// · · · // P1

φ1 // P0

φ0 // M // 0.

It is called infinity if none of Mi is projective.

The following theorem, the ‘Auslander-Buchsbaum formula’, is an effective in-

strument for the computation of the depth of a module.

Theorem 2.3.7: (Auslander-Buchsbaum) Let (R,m, k) be a Noetherian local

ring, and M 6= 0 a finitely generated R-module. If projdim(M) <∞, then

projdim(M) + depth(M) = depth(R).

�

See Appendix A for a discussion of graded case.

Buchsbaum rings enjoy many similar properties of Cohen-Macaulay rings. As a

preparation, we recall some necessary results and definitions.


The following definitions can be found in J. Stuckrad and W. Vogel’s book

“Buchsbaum Rings and Applications”[41].

Definition 2.3.8: Let (R,m) be a local ring. A system of elements f1, · · · , fnof R is called a weak-regular sequence if

m[(f1, · · · , fi−1) : fi] ⊆ (f1, · · · , fi−1), for all i = 1, · · · , n.

�

Definition 2.3.9: Let (R,m, k) be a local ring. Let R be a finitely generated

R-module. We call R a Buchsbaum ring if every system of parameters of R is a

weak-regular sequence. For an arbitrary ring R, R is called Buchsbaum ring if Rm

is a Buchsbaum ring for all maximal ideals m of R �

J. Stuckrad and W. Vogel also expanded the Buchsbaum ring to graded case.

Definition 2.3.10: Let (R,m) be a positively graded ring. R is called a

Buchsbaum ring if every homogeneous system of parameters of R is a weak R regular

sequence.

Over the years, Buchsbaum rings have been one of the main research topics in

commutative algebra and they can be characterized in different ways. Following

criterion is introduced by Trung [45].

Theorem 2.3.11: Let S be a simplicial affine semigroup with spanning vectors

α1, · · · , αm. rank(S) = m. Then R = k[S] is a Buchsbaum ring if and only if

{ω ∈ G(S) |ω + 2αi, ω + 2αj ∈ S for some i 6= j, 1 6 i, j 6 m}+ Hilb(S) ⊆ S.

Example 2.3.12: Let S ⊆ N2 be an affine semigroup defined as S = {(a1, a2) ∈N2 | a1 + a2 ≡ 0 mod 3}. Set S1 = S \ {(2, 1)}. Easy to see that the semigroup ring

k[S1] is a Buchsbaum ring by above theorem.


Definition 2.3.13: Let (R,m, k) be a Noetherian local ring, and M a finite

non-zero R-module of depth t. The number r(M) = dimk ExttR(k,M) is called the

type of M .

Definition 2.3.14: Let R be a Cohen-Macaulay local ring. A maximal Cohen-

Macaulay module C of type 1 and of finite injective dimension is called a canonical

module of R.

Let R be an arbitrary Cohen-Macaulay ring. A finitely generated R-module M

is a canonical module of R if Mm is a canonical module of Rm for all maximal ideals

m of R. �

For a graded ring R, the canonical module in the category of graded R-module

is defined in Chapter 5.

2.3.2. Socle and Type of Positively Graded Rings.

Definition 2.3.15: Let (R,m) be a positively graded ring, and M a finitely

generated non-zero graded R-module of depth t. The number dimR/mExttR(R/m,M)

is called the type of M , denoted by type(M). �

Definition 2.3.16: Let M be a graded module over a positively graded ring

(R,m). Then

Soc(M) = (0 : m)M ∼= HomR(R/m,M)

is called the socle of M . �

Remark: Since Ext0R(k,M) = HomR(k,M) for a finitely generated R-module

M , the type of M of depth 0 is the dimension of its socle.

Proposition 2.3.17: Let (R,m) be a positively graded ring, M a finitely

generated graded R-module and f a maximal homogeneous M -sequence in m. Then

type(M) = dimk Soc(M/fM)

�


The following lemma gives us a hint for proving the proposition.

Lemma 2.3.18: Let R be a ring, M , N be R-modules, and f = f1, · · · , fn a

weak M -sequence in AnnN . Then

HomR(N,M/fM) ∼= ExtnR(N,M)

�

Proof of the Proposition 2.3.17: Let f = f1, · · · , fn be a maximal M -

sequence. Then depth(M) = n.

By the lemma above, we have

HomR(k,M/fM) ∼= ExtnR(k,M)

So

type(M) = dimk ExtnR(k,M) = dimk HomR(k,M/xM) = dimk Soc(M/xM)

�

2.4. Normal and Seminormal Semigroups and Semigroup Rings

2.4.1. Normal Semigroups and Normal Semigroup Rings.

Definition 2.4.1: An affine semigroup S with quotient group G(S) is called

normal if α ∈ G(S) and dα ∈ S imply α ∈ S, where d ∈ N \ {0}.

Easy examples like Nn and S = {(a1, a2) ∈ N2 | a1 + a2 ≡ 0 mod 3} are normal

semigroups.

The proposition below shows that normal affine semigroups can also be defined

by finitely generated rational cones. It will be more convenient to use this proposition

to verify normal affine semigroups.


Proposition 2.4.2: (Gordan’s Lemma) An affine semigroup S is normal if and

only if S = G(S) ∩ C(S).

Let S be an arbitrary semigroup. We shall define the normalization of S as

S = {α ∈ G(S) | dα ∈ S for some d ∈ Z>0}

A domain is normal if it is Noetherian and integrally closed in its fraction field.

We shall see normal semigroups and normal semigroup rings are consistent.

Proposition 2.4.3: (Hochster [24], Proposition 1) Let S be an affine semi-

group. Then S is normal if and only if k[S] is normal.

Theorem 2.4.4: (Hochster [24], Theorem 1) Let S be a normal semigroup.

Then k[S] is Cohen-Macaulay.

2.4.2. Seminormal Semigroups and Seminormal Semigroup Rings.

Definition 2.4.5: A commutative ring R is seminormal if it is reduced and

whenever f, g ∈ R satisfy f 3 = g2, there is an h ∈ R with h2 = f, h3 = g.

A series of parallel results are given for seminormal semigroups and seminormal

semigroup rings.

Definition 2.4.6: A semigroup S is seminormal if α ∈ G(S) and 2α, 3α ∈ S

imply α ∈ S. Such an α is called seminormal.

So the seminormalization of an affine semigroup S, denoted by +S, can be ob-

tained from S by repeatedly adjoining elements α such that 2α and 3α are already

there.

Theorem 2.4.7: (Gubeladze [20], Corollary 3.2) Let S be a positive affine

semigroup, and k a field. Then R = k[S] is seminormal if and only if S is seminormal.

�


The following theorem can be found in [20] by Gubeladze, [43] by Swan, or [32]

By Reid and Roberts.

Theorem 2.4.8: (Reid and Roberts [32], Theorem 4.3) Let S be an affine

subsemigroup of Nn. Then S is seminormal if and only if S =⋃F [(G(S ∩ F ))∩

◦F ]

where F ranges over all faces of C(S).

CHAPTER 3

The Cohen-Macaulay Property of Semigroup Rings

Cohen-Macaulay rings play one of the central roles in present-day Commutative

Algebra. Many algebraic criteria have been given to determine whether a ring is

Cohen-Macaulay. This chapter is organized in two sections. In the first section,

we shall explore the Cohen-Macaulay property of a simplicial affine semigroup ring

which contains all interior lattice points of G(S)∩C(S) in a visual way (well, at least

in a less abstract way), and give a closer answer to problem 10 posted by Bruns,

Gubeladze and Trung [6], which is listed below.

Problem 10: Find Criteria for an affine semigroup ring R = k[S] to be a

Cohen-Macaulay ring in terms of the Hilbert basis of the affine semigroup S.

More precisely, we can determine the Cohen-Macaulay property of a simplicial affine

semigroup ring by checking the cardinality of the spanning monomial set of the

corresponding semigroup.

In the second section, topics on type and socle of Cohen-Macaulay affine semi-

group rings are discussed.

3.1. The Cohen-Macaulay Property of Semigroup Rings

Let S ⊆ Nn be an affine semigroup with a set of generators α1, · · · , αm, β1, · · · , βl ∈S, where α1, · · · , αm are spanning vectors of C(S). We write W for the subsemi-

group of S generated by spanning vectors α1, · · · , αm. Note that α1, · · · , αm may

not be linearly independent. Take fi = xαi , and gj = xβj , i = 1, · · · ,m, j = 1, · · · , l.29


Let R = k[S] = k[f1, · · · , fm, g1, · · · , gl]. B = k[W ] = k[f1, · · · , fm]. In fact R is a

finitely generated B-algebra.

Lemma 3.1.1: R is integral over B.

Proof: It is enough to prove that all monomials in the k-basis of R are integral

over B.

Take any monomial f = xα ∈ R, α ∈ S. Then α can be written as a Q-linear

combination of a subset of α1, · · · , αm by Proposition 2.1.7.

α =∑j∈J

λijαij , λij ∈ Q+, J ⊆ I := [1,m].

Take λi = cibi, ci, bi ∈ N, bi 6= 0, i = 1, 2, · · · ,m. It follows that

α =c1b1α1 +

c2b2α2 + · · ·+ cm

bmαm.

This is equivalent to,

rα = r1α1 + r2α2 + · · ·+ rmαm,

where r =∏m

i=1 bi, ri = rcibi∈ N. Then we have

xrα = xr1α1xr2α2 · · ·xrmαm ,

i.e., f r = f r11 · · · f rmm ∈ B. Therefore, f is integral over B. Hence the lemma follows.

�

This lemma implies that R is a finitely generated B-module. R/(f1, · · · , fm)R

has a unique monomial basis (as k-vector space) with the exponent set, denoted by

T .

In fact, from the semigroup point of view,

T = {t ∈ S | t cannot be written as the form t = ω + α, with ω ∈ W \ {0}, α ∈ S}= {t ∈ S | t− ω /∈ S, for all ω ∈ W \ {0}}= {t ∈ S | t− αi /∈ S, i ∈ I}


so that every element α ∈ S can be written as a sum of one element t in T and one

element in W , i.e.,

α = t+r∑i=1

λiαi, t ∈ T and λi ∈ N for all i.

Such a set T of points is called the spanning set of the semigroup S over the semi-

group W .

Lemma 3.1.2: A minimal set of generators of R as a B-module can be chosen

as the set of monomials XT .

Proof: Since XT is a k-basis of the vector space R/(f1 · · · , fm)R, then we may

take Γ = {f + (f1 · · · , fm)R | for all f ∈ XT}, and clearly Γ is a minimal set of

generators of R/(f1 · · · , fm)R as a k = B/(f1 · · · , fm)B-module.

Let N =∑

f∈XT fB. N is a B-module which is spanned by XT over B. Obvi-

ously R = N +(f1 · · · , fm)R. By graded Nakayama’s Lemma (Atiyah and Macdon-

ald [4]), R = N , i.e., R is spanned by XT as a B-module. XT is minimal since Γ is

minimal.

Hence a minimal generating set of R over B can be chosen as the set of monomials

XT . �

Definition 3.1.3: The monomial subset XT of R is called the spanning mono-

mial set of R as a B-module.

Recall that a simplicial affine semigroup S is a semigroup such that the cone

C(S) has linearly independent spanning vectors α1, · · · , αm. Then in a simplicial

semigroup S, by Box we mean the subset of S such that

Box = {α ∈ G(S) | α =m∑i=1

λiαi, 0 6 λi 6 1, λi ∈ Q}.

Observe that both Box and T lie in the area near the origin of the cone. It is natural

to consider the common elements in both T and Box.


Lemma 3.1.4: Let S ⊆ Nn be a simplicial affine semigroup. Let spanning

vectors of the cone C(S) be α1, · · · , αn, and rank(S) = n. Let the index of the

group G(S) in Zn be h. If S contains all points of G(S)∩◦

C(S), there are at least

| det(A)|/h elements of T in Box, where A =

α1...αn

.

Proof: Define

Box′ = {ω ∈ G(S) | ω =n∑i=1

λiαi, 0 6 λi < 1},

and

Int(Box) = Int(Box′) = {ω ∈ S | ω =n∑i=1

λiαi, 0 < λi < 1}.

Obviously Int(Box′) ⊂ T .

Claim: Box′ contains | det(A) |/h elements of G(S).

Proof: The matrix A represents a map A : Zn → Zn. coker(A) =

Zn/im(A). Hence #coker(A) = | det(A)|.Now let’s consider the group G(S) ⊆ Zn. Note that the image

of A is contained inG(S). Take φ = A. Then the following diagram

commutes.

Zn A //

∼=��

Zn // coker(A) // 0

Znφ // G(S)

⊆

OO

π // coker(φ)

⊆

OO

// 0

The index of G(S) in Zn is h, hence the index of coker(φ) in

coker(A) is h. Therefore #coker(φ) = | det(A)|/h. Note that π

maps Box′ bijectively onto coker(φ). Therefore #Box′ = | det(A)|/h.

Now we build a map ρ from Box′ to T∩Box as follows. The interior of Box′ is

mapped to T∩Box injectively, and for each element ω =∑n

i=1 λiαi ∈ Box′\Int(Box′),

first we lift it by substituting 1 for 0 in the coordinates of ω. This element is in

the interior of C(S), so is in S by the hypothesis. Then we keep subtracting αi’s


corresponding to those λi = 1 in this element until no more αi can be subtracted

in order for it being S. Notice that different choices of what to subtract might

yield different values. Here an arbitrary choice is being made, and the point must

be in both T and Box. Now we have a map ρ : Box′ → T ∩ Box. Obviously

the map of all points in Int(Box′) is 1-1, and since for different elements in (T ∩Box)\int(Box), when we recover them to Box′\Int(Box′), they are always different,

ρ is an injection. Therefore T ∩ Box contains at least | det(A)|/h elements. Hence

the lemma follows. �

In fact, Lemma 3.1.4 is true for a more general case in which the rank m of S is

less than n. Let V denote the Q-linear subspace generated by the spanning vectors

of the cone C(S). Under the above condition,

G(S) ⊆ Zn ∩ V ∼= Zm =: Zm.

Let the index of G(S) in this particular Zm, or Zm, be h.

Now take the spanning vectors α1, · · · , αm of C(S) as the rows of the matrix A,

i.e.,

A =

α1...αm

.

Then A represents a homomorphism φ : Qm → Qn given by φ(α) = αA for α ∈ Qm.

Clearly there is a diagonal matrix

D =

a1 0 · · · 0 0 · · · 00 a2 · · · 0 0 · · · 0

· · · · · ·0 0 · · · am 0 · · · 0

m×n

where 0 < a1 6 · · · 6 am, ai ∈ Z, for all i

such that the following diagram

Qm A // Qn

Q

��Qm D //

P

OO

Qn

(1)


commutes and D = PAQ, where P and Q are chosen as an invertible m × m

matrix and an invertible n × n matrix with integer entries, respectively. Define

det(A) = det(D) := a1 · · · am.

The following lemma holds.

Lemma 3.1.5: Let S ⊆ Nn ∩ C(S) be a simplicial affine semigroup. Suppose

that the spanning vectors of C(S) are α1, · · · , αm, and rank(S) = m 6 n. Let the

index of the group G(S) in Zm be h. If S contains all points of G(S)∩◦

C(S), there

are at least | det(A)|/h elements of T in Box.

Proof: Since A, P , Q, and D have integer entries in diagram (1), the diagram

Zm A // Zn

Q

��Zm D //

P

OO

Zn

commutes as well.

Let π : Zn → Zm be the projection mapping the elements in Zn to the first m

coordinates. Define

D′ =

a1 0 · · · 00 a2 · · · 0

· · ·0 0 · · · am

m×m

Then the following diagram commutes,

Zm A // Zn

Q��

Zm

P

OO

D //

D′&&MMMMMMMMMMMM Zm ⊕ Zn−m

π

��Zm

Since rank(S) = m 6 n, the semigroup S is isomorphic to a semigroup U = πQ(S)

in Zm with spanning vectors of C(U) the rows of P−1D′. It is easy to show that for


every element ω ∈ S, it is in the spanning set T of S if and only if the image of ω

under the map πQ, i.e., (ω)πQ, is in the spanning set TU of U .

Note that coordinates of rows of P−1D′ may not be positive. Thanks to the uni-

modular transformation, the semigroup U can be transformed isomorphically onto

the first orthant of Qn. Therefore we may assume U ⊆ Nn and use previous results

without any difficulties. By Lemma 3.1.4, Box of U contains at least | det(D′)|/helements of the spanning set TU of U . Simply lifting these elements up to Zn, we

sent them to the spanning set T of S which is also in Box of S. Hence Box of S

contains at least | det(D′)|/h = | det(A)|/h elements. �

Let we see an example first.

Example 3.1.6: Let S ⊆ N2 be the affine semigroup with Hilbert basis

Hilb(S) = {(6, 0), (5, 1), (2, 1), (1, 2), (1, 5), (0, 6)}.

The spanning vectors of C(S) are α1 = (6, 0), α2 = (0, 6). det(A) =

(6 00 6

)= 36.

R = k[S] = k[x6, x5y, x2y, xy2, xy5, y6] is a Cohen-Macaulay ring. (This is obtained

by using the methods presented in Appendix A).

The group generated by S is G(S) = {(x, y) ∈ Z2 |x+ y ≡ 0 mod 3}. It is easy

to show that the index of G(S) in Z2 is 3.

The spanning set T is illustrated by the big dots in the following figure. #T =

| det(A)|/3 = 12

-

6

b b b b b b b b bb b b b b b b b bb b b b b b b b bb b b b b b b b bb b b b b b b b bb b b b b b b b bb b b b b b b b bb b b b b b b b b

r rr r rr r rr rr r rr r rr r rr r r

t tt tt tt tt tt

(0,0)

tα1

α2 S

�


In the example above, the equality #T = | det(A)|/h is not a coincidence. In

fact, the number | det(A)|/h is so important that it indicates the Cohen-Macaulay

property of a simplicial affine semigroup ring.

Let’s start from a special case in which the rank of S is n.

Theorem 3.1.7: Let S ⊆ Nn be a simplicial affine semigroup withG(S)∩◦

C(S)⊂S. Let the spanning vectors of the cone C(S) be α1, · · · , αn. Suppose that the index

of the group G(S) in Zn is h. Then R = k[S] is a Cohen-Macaulay ring if and only

if the cardinality of the spanning monomial set XT is | det(A)|/h. �

To prove this theorem, we need the following lemmas.

Lemma 3.1.8: Let G be a subgroup of Zn, and the index of G in Zn be h. Let

M , N denote the group rings k[Zn], k[G] respectively. Then the group ring M is a

free N -module of rank h.

Proof: By the hypothesis (Zn : G) = h, Zn is a disjoint union of h cosets, i.e.,

Zn = ∪h−1i=0 (zi + G), where z0 = 0, zi ∈ Zn for i = 1, · · · , (h − 1). This implies that

for any z ∈ Zn, z can be uniquely written as zi + g with 0 6 i 6 h − 1 and g ∈ G.

Therefore every monomial xz in the k-basis of the group ring M = k[Zn] can be

expressed as xz = xzixg uniquely, where xzi ∈ M,xg ∈ N = k[G]. So xz0 , · · · , xzh−1

generate the group ring M as an N -module.

To see that M is a free N -module with rank h, it is enough to prove that the

above generating set {xz0 , · · · , xzh−1} is linearly independent.

Suppose there are elements w0, · · · , wh−1 ∈ N such that

w0xz0 + w1x

z1 + wh−1xzh−1 = 0

Each term wixzi contains monomials in the set {xz ∈M | z ∈ zi +G}, moreover the

monomials xz are linearly independent over k. Therefore every term wixzi is equal

to zero. Note that the group ring M is a domain, so wi = 0 for all i. �


Lemma 3.1.9: Let R and B be finitely generated domains with B ⊆ R. Let

K, L be fraction fields of R and B respectively. Then R is a free B-module if and

only if R can be generated as a B-module by [K : L] elements.

Proof: “⇐” Let Γ denote a generating set of R over B with #Γ = [K : L]. It

is enough to show that Γ is linearly independent.

Since L is the fraction field of B, then L = B∆ where ∆ = B \ {0}. We have

L⊗B R = B∆ ⊗B R = R∆ ⊆ K.

But R∆ is an integral domain and K is integral over L⊗BR = R∆, so by [4] (Atiyah

and Macdonald) Proposition 5.7, L⊗B R is a field. Hence K ⊆ L⊗B R. Therefore

L⊗B R = K.

By the assumption that Γ generates R as a B-module, then Γ generates L⊗B R

over L ⊗B B, i.e., it spans K over L. But #Γ = [K : L]. Hence Γ is linearly

independent over L, so is over B.

“⇒” SinceR is a freeB-module, then the cardinality of the free basis is rankBR =

[K : L]. �

Lemma 3.1.10: Let S ⊆ Zn be a simplicial affine semigroup with rank n. Let

α1, · · · , αn denote the spanning vectors of the cone C(S). R = k[S]. B = k[W ],

where W denotes the semigroup generated by α1, · · · , αn. Let K and L be the

fraction fields of R and B respectively. Then [K : L] = | det(A)|/h. where A = α1...αn

.

Proof: In the diagram (1) on page 33, since A, D, P , and Q have integer entries,

the diagram

Zn A // Zn

Q

��Zn D //

P

OO

Zn


commutes with

D = PAQ =

a1 0 · · · 00 a2 · · · 0

· · ·0 0 · · · an

.

In fact, W = im(A|Nn).

From the fact that the mapping Zn → k[x1, x−11 , · · · , xn, x−1

n ] given by α 7→ xα

provides an isomorphism between the group Zn and the multiplicative group of

monic monomials of the Laurent polynomial ring k[x1, x−11 , · · · , xn, x−1

n ], the homo-

morphism φ : Zn → Zn represented by a matrix M induces a ring homomorphism

φ : k[t1, t−11 , · · · , tn, t−1

n ] → k[x1, x−11 , · · · , xn, x−1

n ] which is given by φ(tα) = tαM =

xφ(α) = xαM for any monomials tα ∈ k[t1, t−11 , · · · , tn, t−1

n ]. Then the above diagram

induces the following commutative diagram of Laurent polynomial rings,

k[t1, t−11 , · · · , tn, t−1

n ]A // k[x1, x

−11 , · · · , xn, x−1

n ]

∼= Q��

k[t1, t−11 , · · · , tn, t−1

n ]

∼=P

OO

D // k[x1, x−11 , · · · , xn, x−1

n ]

such that D = P AQ. Then for each tα ∈ k[t1, t−11 , · · · , tn, t−1

n ], α ∈ Zn,

tαP AQ = tαD = tαD = tαPAQ = tαPAQ

Therefore, the composition of the homomorphisms can be interpreted by the prod-

uct of the matrices. Obviously Q induces an isomorphism im(A) ∼= im(D) =

k[xa11 , x

−a11 , · · · , xan

n , x−ann ]. Furthermore QmapsB = k[W ] ⊆ im(A) into k(xa1

1 , · · · , xann ).

Then L, the fraction field of B, is mapped isomorphically via Q to the fraction field

of im(D), i.e., k(xa11 , · · · , xan

n ).

Now we have L ⊆ K ⊆ H = k(x1, · · · , xn). So [H : L] = [H : K][K : L]. Clearly

[H : L] = | det(D)| = |a1 · · · an| = | det(A)|. By applying Lemma 3.1.8 and Lemma

3.1.9,

[H : K] = [ fraction field of k[Zn] : fraction field of k[G(S)] ]= [ Zn : G(S) ]= h


Hence [K : L] = | det(A)|/h. �

Theorem 3.1.7 now is a consequence of Lemma 3.1.9, Lemma 3.1.10, and the

theorem of H. Derksen and G. Kemper [9] (Theorem 2.3.6 in this thesis). �

We are hoping that Theorem 3.1.7 is also true when m < n. By analyzing the

proof of Theorem 3.1.7, we find that the only gap we need to fill in is Lemma 3.1.10.

The next Lemma is an alternative version of Lemma 3.1.10.

Lemma 3.1.11: Let S ⊆ Nn∩C be a simplicial affine semigroup. Suppose the

spanning vectors of the cone C(S) are α1, · · · , αm. rank(S) = m 6 n. Then there is

an affine semigroup S∗ ⊆ Zm ∩ C(S∗) such that G(S) ∼= G(S∗), the subrings R∗ =

k[S∗] and B∗ of k[x±1 , · · · , x±m] are compatibly isomorphic to R and B respectively,

and the degree of their field extensions is | det(A)|/h, where A and det(A) are defined

as before.

Proof: Using the same notation A, D, P , and Q as before, we have the following

diagram.

Zm A // Zn

Q

��Zm D //

P

OO

Zn

(2)

Clearly, S ⊂ im(A) in the diagram (1) on page 33,

W = {ω ∈ Nn |ω =m∑i=1

λiαi, λi ∈ N for all i} = im(A|Nm)

in the diagram (2), and αiQ = (ai1, · · · , aim, 0, · · · , 0) := (α∗i , 0), α∗i ∈ Zm for

all i. The linear independence of α1, · · · , αm implies that α∗1, · · · , α∗m are linearly

independent.


Define a projection π : Qn → Qm by (s1, · · · , sm, sm+1, · · · , sn) 7→ (s1, · · · , sm).

Let

D∗ =

a1

a2

· · ·am

We have the following commutative diagram,

Qm A // Qn

Q��

Qm

P

OO

D //

D∗&&MMMMMMMMMMMM Qm ⊕ Qn−m

π

��Qm

Equivalently, the diagram

Qm AQπ // Qm

Id��

Qm

P

OO

D∗// Qm

(3)

commutes, where Id is the m×m identity matrix.

Let S∗ = (S)Qπ ⊆ Zm. Then Qπ : S → S∗ is surjective. Since Q : Qn →Qm⊕Qn−m maps S into the first summand, Qπ|S is injective as well. SoQπ : S → S∗

is an isomorphism. It follows immediately that S∗ is a semigroup in Zm, i.e., Zm,

generated by α∗1, · · · , α∗m, and G(S) ∼= G(S∗). Note that we may assume that

S∗ ⊆ Nn by the unimodular Transformation (on page 16).

Let R∗ = k[S∗]. Then R ∼= R∗.

By the diagram (3), we have the following commutative diagram

Zm AQπ // Zm

Id��

Zm

P

OO

D∗// Zm

(4)

as all the entries of the matrices are integers.


Let W ∗ be the semigroup generated by α∗1, · · · , α∗m in Zm. In fact W ∗ = (W )Qπ.

Define B∗ = k[W ∗]. Then B∗ ∼= B. Clearly W ∗ is a subsemigroup of S∗, B∗ is a

subring of R∗. It is easy to see that R∗ is a finitely generated B∗ module, and the

diagram

R∼= // R∗

B∼= //

OO

B∗

OO

commutes. So the spanning monomial set XT of R over B is isomorphic to the

spanning monomial set XT ∗ of R∗ over B∗.

Take A∗ =

α∗1...α∗m

= AQπ. Then we have the commutative diagram

Zm A∗// Zm

Id��

Zm

P

OO

D∗// Zm

Let K∗, L∗ be the fraction fields of R∗ and B∗ respectively. By Lemma 3.1.10,

[K∗ : L∗] =| det(D∗)|

the index of G(S∗) in Zm = | det(A)|/h.

where det(A) is defined to be det(D∗) = a1 · · · am. �

Then we have the general version of Theorem 3.1.7.

Theorem 3.1.12: Let S ⊆ Nn ∩ C be a simplicial affine semigroup with

G(S)∩◦

C(S)⊂ S. Let the spanning vectors of the cone C(S) be α1, · · · , αm.

rank(S) = m 6 n. Let the index of the group G(S) in Zm be h. Then R is

Cohen-Macaulay if and only if the cardinality of the spanning monomial set XT is

det(A)/h. �

By the above theorem and Lemma 3.1.5, we have the following result.


Theorem 3.1.13: Let S ⊆ Nn∩C(S) be a simplicial affine semigroup contain-

ing all points of G(S)∩◦

C(S). Let the index of the group G(S) in Zm be h. If the

spanning set T of S contains elements outside of the Box, R is not Cohen-Macaulay.

�

Rosales and Garcıa-Sanchez studied the spanning set of a simplicial affine semi-

group in their paper [36]. They considered the intersection of the Apery sets, i.e.,

∩S(α), of the simplicial semigroup S, and characterized Cohen Macaulayness by

using this intersection (Corollary 1.6, [36]).

Recall the Apery set of an element α 6= 0 in a semigroup S are defined as the set

S(α) = {ω ∈ S |ω − α /∈ S}.

Clearly the spanning set T we are using is nothing but the intersection of the Apery

sets of the spanning vectors α1, · · · , αm on the extreme rays of the semigroup S, i.e.,

T =m⋂i=1

S(αi).

They also defined Box of an affine semigroup S (page 526, [36]).

Theorem 1.5 of [36] is the same as the equivalence of (1) and (4) in Derksen

and Kemper’s theorem [9] (Theorem 2.3.6 in this thesis). They did not explicitly

introduce R and B, but express their result in a form equivalent to freeness of R as

a B-module. Furthermore they didn’t develop the criterion of Theorem 3.1.12.

Let S ⊆ Nn be an affine semigroup with rank r 6 n, and let the spanning

vectors of the cone C(S) be α1, · · · , αm. Let F be a proper d-dimensional face of

C(S) spanned by a subset ΓF of the set of spanning vectors, denoted by Γ, of S.

Let WF be the semigroup generated by ΓF . Take RF = k[S ∩ F ] and BF = k[WF ].

Then RF is a finitely generated BF -module with spanning monomial set denoted by

XTF . Equivalently, TF consists of all elements in S ∩ F which cannot be written as

the form β + ω with ω ∈ WF \ {0} and β ∈ S ∩ F .


Lemma 3.1.14: TF = T ∩ F .

Proof: With no loss of generality, we may assume that F is spanned by α1, · · · , αd,where d < m.

TF ⊆ T ∩ F . If not, there is a tF ∈ TF ⊂ S ∩ F , but tF /∈ T ∩ F . This

implies tF /∈ T because tF ∈ F . Then tF = t +∑m

i=1 λiαi, where t ∈ T , λi ∈ N for

i = 1, · · · ,m, and not all λis’ are zero.

Suppose t =∑m

i=1 µiαi, µi ∈ Q>0. Therefore

tF =∑m

i=1 µiαi +∑m

i=1 λiαi=

∑mi=1(µi + λi)αi

Note that tF ∈ S∩F . Then µi+λi = 0 for i = (d+1), · · · ,m. This implies µi = λi =

0 for i = (d+ 1), · · · ,m since they are non-negative. Hence t =∑d

i=1 µiαi ∈ S ∩ Fand tF = t+

∑di=1 λiαi, where

∑di=1 λiαi ∈ WF . So tF /∈ TF . A contradiction.

For the other direction assume there is a t ∈ T ∩ F , but t /∈ TF . Then t = tF +∑di=1 λiαi, where λi ∈ N, some λi’s are not zero, tF ∈ TF ⊂ S, and

∑di=1 λiαi ∈ W .

So t /∈ T . A contradiction. �

Example 3.1.15: Let the Hilbert basis of a affine semigroup S be the following,

Hilb(S) = {(2, 0, 0), (0, 2, 0), (0, 0, 3), (1, 1, 0), (1, 0, 1), (0, 1, 1), (1, 1, 1)}

C(S) = N3. The spanning vectors of C(S) are α1 = (2, 0, 0), α2 = (0, 2, 0), α3 =

(0, 0, 3). Take a 2-dimensional face F of C(S) to be the yz coordinate plane. Then

R = k[x2, y2, z3, xy, xz, yz, xyz], and RF = k[y2, z3, yz].

The following computer script shows that TF = T ∩ F by using the computer

program in Appendix B.

In[1]:= <<’’A:\\TotalProgram.m’’

In[2]:= A = {{2, 0, 0}, {0, 2, 0}, {0, 0, 3}}

Out[2]= {{2, 0, 0}, {0, 2, 0}, {0, 0, 3}}

In[3]:= S = {{2, 0, 0}, {0, 2, 0}, {0, 0, 3}, {0, 1, 1},


{1, 0, 1}, {1, 1, 0}, {1, 1, 1}}

Out[3]= {{2, 0, 0}, {0, 2, 0}, {0, 0, 3}, {0, 1, 1},

{1, 0, 1}, {1, 1, 0}, {1, 1, 1}}

In[4]:= T= basis[S, A]

Out[4]= {{0, 0, 0}, {0, 1, 1}, {1, 0, 1}, {1, 1, 0}, {1, 1, 1},

{0, 2, 2}, {1, 1, 2}, {1, 2, 2}, {2, 0, 2}, {2, 1, 2},

{2, 2, 1}, {0, 3, 3}, {1, 2, 3}, {2, 1, 3}, {3, 0, 3},

{0, 4, 4}, {4, 0, 4}, {0, 5, 5}, {5, 0, 5}}

In[5]:= AF={{0,2,0},{0,0,3}}

Out[5]= {{0,2,0},{0,0,3}}

In[6]:= SF={{0,2,0},{0,0,3},{0,1,1}}

Out[6]= {{0,2,0},{0,0,3},{0,1,1}}

In[7]= TF= basis[SF,BF]

Out[7]= {{0, 0, 0}, {0, 1, 1}, {0, 2, 2},

{0, 3, 3}, {0, 4, 4}, {0, 5, 5}}

F is the yz coordinate plane. In fact, the intersection of F and

T is the set of points in T whose first coordinates are 0.

In[8]:= Select[T, #[[1]]==0&]

Out[8]= {{0, 0, 0}, {0, 1, 1}, {0, 2, 2},

{0, 3, 3}, {0, 4, 4}, {0, 5, 5}}

�

Very often we shall consider the properties of the spanning monomial set of a

finitely generated module over a semigroup ring. The following lemma is basic for

such arguments.

Lemma 3.1.16: Let S ∈ Nn be an affine semigroup, W be a subsemigroup of

S. Take R = k[S] and B = k[W ]. Let R be a finitely generated B-module with the


spanning monomial set XT . If R is not a free B-module, there must exist v ∈ S

such that v = t+ ω = t′ + ω′ where t 6= t′ ∈ T , ω, ω′ ∈ W .

Proof: We know that R is a finitely generated B-module, but not free. so there

exists a f ∈ R with at least two different expressions, i.e.,

f =∑i

λixtixωi =

∑j

µjxtjxωj ,

where xωi , xωj ∈ B, λi, µj ∈ k, and ti, tj ranging over T .

Without loss of generality, we may assume that f is homogeneous of degree v.

Since these expressions are different, there exists i such that λi 6= µi. In order

to cancel the term xtixωi , there must be ti 6= tj such that xv = xtixωi = xtjxωj ,

equivalently, v = ti + ωi = tj + ωj, with ti 6= tj. �

Based on these two lemmas above, we have the following theorem.

Theorem 3.1.17: RF = k[S ∩ F ] is Cohen-Macaulay if R = k[S] is Cohen-

Macaulay.

Proof: Suppose not. This is equivalent to say that RF is not a free BF module.

Then by Lemma 3.1.16, there is a ν ∈ S ∩F such that ν = t1 + ω1 = t2 + ω2, where

t1 6= t2 ∈ TF , ω1, ω2 ∈ WF . But TF ⊂ T , WF ⊂ W , and S ∩F ⊂ S. So ν ∈ S can be

written in two different forms t1 +ω1 and t2 +ω2 with t1, t2 ∈ T and ω1, ω2 ∈ W . It

follows immediately that R is not a free B module, i.e., R is not Cohen-Macaulay.

A contradiction. �

Remark: 1. With notation the same as above, suppose WF is generated by

ΓF = {αi1 , · · · , αid}, a subset of the spanning vectors Γ of C(S). In general,

RF 6= R/(xαid+1 , · · · , xαim )R

where {αid+1, · · · , αim} = Γ \ ΓF .

Example 3.1.18: R = k[x2, y2, z2, xyz, x2yz, xy2z, xyz2]. The spanning vec-

tors of the corresponding semigroup S are α1 = (2, 0, 0), α2 = (0, 2, 0), α3 = (0, 0, 2).


Pick a face F of C(S), for example, the xy coordinate plane. Then RF = k[x2, y2].

Clearly, R/z2R 6= RF . �

Remark: 2. The converse of this theorem is not true, i.e., even if all rings

associated with proper faces of S are Cohen-Macaulay, R is not necessarily Cohen-

Macaulay.

Example 3.1.19: R = k[S] = k[x, y, z2, xz, yz]. The affine semigroup rings of

the proper faces of S are k[x], k[y], k[z2], k[x, y], k[x, z2, xz], k[y, z2, yz]. They are

all Cohen-Macaulay rings by Theorem 3.1.7. But R is not. �

3.2. Socle and Type of a Special Class of Cohen-Macaulay Semigroup

Rings

In this section we shall deal with the following case:

Let S ⊆ Nn be a simplicial affine semigroup with rank m 6 n. Suppose spanning

vectors of C(S) are α1, · · · , αm. Recall that W denotes the subsemigroup of S

generated by these spanning vectors. Suppose that S consists of all the interior

integer points of the cone C(S), i.e.,◦

C(S) ∩Nn ⊆ S, and there are no other points

of S on the proper faces of C(S) except the points of W . Then we have the following

theorem.

Proposition 3.2.1: Such an affine semigroup ring R = k[S] is Cohen-Macaulay

with a regular sequence x := xα1 , · · · , xαm .

Proof: By the definition, the spanning set XT is the k-basis of

R/(xα1 , · · · , xαm)R. Obviously #XT = | det(A)|.

It easy to show that the fields of fractions of R and B are K = k(x1, · · · , xm)

and L = k(xα11 , · · · , xαm

n ) respectively. So [K : L] = | det(A)| = #XT . The theorem

holds by Theorem 3.1.12. �


Now let S ⊆ Nn be a standard affine simplicial semigroup with rank(S) = n.

Then C(S) = Qn+, the first orthant in the coordinate space. Hence spanning vectors

of C(S) will be α1 = (a1, 0, · · · , 0), · · · , αn = (0, · · · , 0, an). Suppose◦

C(S) ∩Nn ⊆ S,

and whenever an α ∈ S ∩ F , where F is a proper face of C(S),

α =n∑i=1

λiαi, λi ∈ N, i = 1, · · · , n.

So by the character of R, T \ {0} looks like an n-dimensional cubic polyhedron with

a corner cut in Qn. The subset of lattice points ω = (ω1, · · · , ωn) in T with ωi = ai

or ωi = 1 for some i is called the top or the bottom face of T respectively.

It is easy to see that Hilb(S) consists of the elements (a1, · · · , 0), · · · , (0, · · · , an)on the axes, and the elements on the (n − 1)-dimensional bottom faces of T . In

other words, the generating set of k[S] includes xa11 , · · · , xan

n , and all the elements

xω, ω = (ω1, · · · , ωn) ∈ S, such that at least one ωi is 1 and 1 ≤ ωj ≤ aj, for all

j 6= i. The case n = 2 is illustrated below.

-

6

q q q q qq q q q qq q q q qq q q q qq q q q q

(a1, 0)q

(0, a2) q· · ·

...

(0,0)q

T \ (0, 0)

S

By Proposition 3.2.1, the semigroup ring R = k[S] is Cohen-Macaulay and the

regular sequence of R is x = xα1 , · · · , xαm = xa11 , · · · , xan

n .

Recall that the socle of the quotient ring (R/xR) is the annihilator of maximal

ideal in R/xR. Well, the homogeneous maximal ideal of R/xR is the ideal generated

by XT\{0}. So the socle of the quotient ring R/xR is,

Soc(R/xR) = {y : y ∈ R/xR such that yh ∈ xR, for all h ∈ XT\{0}}.


Evidently, the elements xa11 , · · · , xan

n and xω, ω ∈ S\T are in xR. So the elements

in the k-basis of Soc(R/xR) are those y ∈ R/xR such that the index of yh jumps

out of T for all h ∈ XT\{0}. Actually it is enough to find the elements y ∈ XT such

that y(x1 · · ·xn) ∈ xR. Therefore if none of the ai is 1, the elements in a k-basis of

Soc(R/xR) are those whose index is on the “roof” of T . So we have,

Theorem 3.2.2: If none of ai is 1, then the k basis of socle of R/xR consists

of the element xa1−11 · · ·xan−1

n and all the elements xγ = xγ11 · · ·xγnn such that at least

one γi is ai and 0 < γj ≤ aj for all j 6= i, except the element xa11 · · ·xan

n . �

Example 3.2.3: Let S ⊂ Z2 be an affine semigroup defined as following,

S = {(a1, a2) | ai > 0} ∪ {(3λ1, 0)| λ1 ∈ N} ∪ {(0, 4λ2)| λ2 ∈ N}.

The spanning vectors are α1 = (3, 0), α2 = (0, 4). The semigroup S and T , the index

set of the spanning monomial set of R, are illustrated below.

-

6

q q q q qq q q q qq q q q qq q q q q

(3,0) (6,0)q q

(0,4) q· · ·

...

(0,0)p

T \ (0, 0)

S

The Hilbert basis Hilb(S) of S exactly consists of all the points lying on the

2− 1 = 1 dimensional bottom faces of T and the two points (3, 0), (0, 4). So

R = k[S] = k[x31, x

42, x1x2, x

21x2, x

31x2, x1x

22, x1x

32, x1x

42]

The maximal homogeneous ideal of R/(x31, x

42)R is

m = (x1x2, x21x2, x

31x2, x1x

22, x1x

32, x1x

42).

By definition, a k-basis of Soc(A/(x31, x

42)A) is {x1x

42, x

21x

42, x

21x

32, x

31x

32, x

31x

22, x

31x2}.


In the picture below, the elements in Hilb(S) correspond to the solid dots, and

the elements in k-Soc(A/(x31, x

42)A) are represented by the circles.

-

6

q q qqqqd d

dddd

(3,0)q

(0,4)q· · ·

...

(0,0)p

�

Compute the Type

Let S, R and T be defined as before, x be a regular sequence in R. Then we

have,

Theorem 3.2.4: Let G(t) = (a1 − t) · · · (an − t). If none of the ai is 1, then

type(R) = G(0)−G(1)

Proof: As we have mentioned before, the type of R,

type(R) = dim(Soc(R/xR))

According to the characterization of elements in Soc(R/xR), to compute the

type, it is enough to compute the number of the points on the “roof” of T .

Geometrically, it is same to add the point (a1, · · · , an) instead of the point (a1−1, · · · , an − 1). So let T ′ be a cubic polyhedron mended by adding (a1, · · · , an) into

T \{0}. Let F1, · · · ,Fn denoted the (n−1)-dimensional faces of T ′ whose equations

are xi = ai, i = 1, · · · , n of T ′. Then the type is the number of the integer points

on the faces F1, · · · ,Fn of T ′, i.e.,

type(R) = |F1 ∪ F2 ∪ · · · ∪ Fn|.


Let G(t) = (a1 − t) · · · (an − t), then

G(1) = a1 · · · an −∑

1≤i≤n

a1 · · · ai · · · an +∑

1≤i<j≤n

a1 · · · ai · · · aj · · · an + · · ·+ (−1)n

G(0) = a1 · · · an

Therefore,

type(R) = |F1 ∪ F2 ∪ · · · ∪ Fn| = G(0)−G(1)

�

Example 3.2.6: (1) In Example 3.2.3, a1 = 3, a2 = 4. Then

type(R) = (a1 + a2)− 1 = (3 + 4)− 1 = 6

(2) Let S ⊂ Z3 be an affine standard simplicial semigroup with N3∩◦

C(S)⊂ S.

Let the spanning vectors of the cone C(S) be (3, 0, 0), (0, 4, 0), (0, 0, 3). Assume

there are no other points on facets of the cone except the points of W which is

the semigroup generated by spanning vectors. The type of R can be calculated as

follows,

type(R) = (a1a2 + a1a3 + a2a3)− (a1 + a2 + a3) + 1

= (12 + 9 + 12)− (3 + 4 + 3) + 1

= 24

�

Remark: If at least one ai is 1, then G(1) = 0, the result will be different.

Example 3.2.7: Let S ⊆ Z3 be an affine semigroup defined as before. Let

spanning vectors of the cone be α1 = (1, 0, 0), α2 = (0, 3, 0), α3 = (0, 0, 3). Then

R = k[x1, x32, x

33, x1x2x3, x1x

22x3, x1x

32x3, x1x2x

23, x1x

22x

23, x1x

32x

23, x1x2x

33, x1x

22x

33]

The corresponding Hilbert basis of S are represented by the solid dots below.


r(1,0,0)r(0,3,0)

r(0,0,3)

r r rr r rr r

e e ee e ee e(1,0,3)

(1,3,0)

so the k-basis of the quotient ring R/(x1, x32, x

33)R is

{x1x2x3, x1x22x3, x1x

32x3, x1x2x

23, x1x

22x

23, x1x

32x

23, x1x2x

33, x1x

22x

33}

which is illustrated by the circles in the graph. Therefore type(R) = 8, while

G(0)−G(1) = 9. �

When some ai are 1 but at least one ai is not 1, the socle of R/xR is exactly

XHilb(S)\{x}. Actually to find the type it is enough to calculate the number of points

on only one face of T with the “top” point missing. Hence type(R) = a1 · · · an−1 =

G(0)− 1.

Theorem 3.2.8: If at least one ai is 1 but not all ai = 1, then

Soc(R/xR) = XHilb(S)\{x}

type(R) = G(0)− 1 = a1 · · · an − 1.

�

Remark: If all ai = 1, i.e., k[S] = k[x1, · · · , xn], then type(A) = 1.

Using the observation that the index of elements in XHilb(S)\{x} are opposite to

the index of elements in the k-basis of Soc(R/xR), we now have that

Theorem 3.2.9: Let ai > 1 for all i. If not all ai = 1, then type(R) =

#Hilb(S)− n �

CHAPTER 4

Seminormality of Semigroups and Semigroup Rings

In this chapter, we shall investigate the coincidence between Cohen-Macaulay

property (CM) and seminormality (SN) of affine semigroup rings. As a preparation,

in section 1 we introduce a new practical criterion to indicate the seminormality of

simplicial affine semigroups. We launch the discussion toward the coincidence in

section 2 starting from simplicial affine semigroup rings. Arbitrary affine semigroup

rings are considered in section 3 and 4. A conjecture is given in section 4, and is

proved when the rank of the semigroup r 6 3.

4.1. A New Criterion for Seminormal Semigroups

Let S ⊆ Nn be a simplicial affine semigroup. Let spanning vectors of C(S) be

α1, · · · , αm. In chapter 3 we gave the following definitions of Box and the spanning

set T of S,

Box = {ω ∈ S |ω =m∑i=1

λiαi, 0 6 λi 6 1}

T = {ω ∈ S | ω − αi /∈ S, for all i ∈ I}

What we know about a given affine semigroup is usually its Hilbert basis. Fol-

lowing is a new practical computational criterion for a simplicial affine semigroup S

to be seminormal in terms of its Hilbert basis.

Theorem 4.1.1: A simplicial affine semigroup S is seminormal if and only if

the spanning set T is contained in Box.

53


Proof: “⇒” If S is seminormal, by Theorem 2.4.8, S =⋃F [(G(S ∩ F ))∩

◦F ]

where F ranges over all faces of C(S).

It is enough to show that the condition ω /∈ Box implies ω /∈ T for any ω ∈ S.

Take ω ∈ G(F )∩◦F⊆ S on a d-dimensional face F of S which is generated

by a subset of α1, · · · , αm denoted by αi1 , · · · , αid . Then ω =∑d

j=1 λjαij , where

λj ∈ Q>0 for all j’s. Assume ω /∈ Box. Then there is a λl in the expression for ω

such that λl > 1.

Let’s calculate ω − αl.

ω − αl =∑j 6=l

λjαij + (λl − 1)αl

Note that λj > 0 for all j 6= l and λl−1 > 0. So ω−αl ∈◦F . Moreover ω−αl ∈ G(F )

since ω, αl ∈ G(F ). It follows immediately that ω − αl ∈ G(F )∩◦F . So ω − αl ∈ S.

This implies ω /∈ T .

“⇐” Suppose the semigroup S is not seminormal. Then there is an ω ∈ G(S)

such that 2ω, 3ω ∈ S, but ω /∈ S.

In fact ω must lie in the interior of a face F (including the cone itself) of the

cone C(S) by Caratheodory’s theorem (Ziegler [48], p46). Assume F is spanned by

a subset of α1, · · · , αm, say αi1 , · · · , αid . Then ω =∑d

j=1 λjαij with λj ∈ Q>0.

Claim: There is an integer l > 2 and a (N1, · · · , Nd) ∈ Zd+ such

that

ω +d∑j=1

Njαij = lω ∈ S∩◦F .

Proof: Note that lω ∈ S∩◦F for all integers l > 2 since 2ω, 3ω ∈

S∩◦F . Then by Proposition 2.1.7, for each l > 2, (l − 1)ω can be

written as a unique Q-linear combination of αi1 , · · · , αid , equiva-

lently, there is a (γ1, · · · , γd) ∈ Qd>0 such that (l−1)ω =

∑dj=1 γjαij .

After choosing a suitable multiple, there must be an integer l > 2


and a (N1, · · · , Nd) ∈ Zd>0 such that

∑dj=1Njαij = (l−1)ω ∈ S∩

◦F ,

i.e.,

ω +d∑j=1

Njαij = lω ∈ S∩◦F

Now keep subtracting αil ’s from ω+∑d

j=1Njαij until we find such an (N1, · · · , Nd) ∈Nd that ω +

∑dj=1 Njαij − αil /∈ S for all l, but ω +

∑dj=1 Njαij ∈ S. In fact,

ω +∑d

j=1 Njαij − αr /∈ S for all r ∈ I. Hence ω +∑d

j=1 Njαij ∈ T .

Observe that at least one Nl > 1. So ω +∑d

j=1 Njαij /∈ Box. A contradiction.�

Based on the result presented above, we can actually “calculate” seminormality

of a positive affine semigroup.

Example 4.1.2: It is easy to check that the semigroup

S = N3 \ {(0, 0, c) ∈ N3 | c ≡ 1 mod 2}

associated with R = k[S] = k[x, y, z2, xz, yz] is seminormal by Theorem 2.4.8.

We may take spanning vectors α1 = (1, 0, 0), α2 = (0, 1, 0) and α3 = (0, 0, 2). By

the definition, Box of S contains points

(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0), (1, 0, 1), (0, 1, 1)(1, 1, 1), (0, 0, 2), (1, 0, 2), (0, 1, 2), (1, 1, 2)

According to the computer program in appendix B, the spanning set is

T = {(0, 0, 0), (1, 0, 1), (0, 1, 1)} ⊆ Box.

�

Example 4.1.3: Let S be a simplicial affine semigroup with Hilbert basis

Hilb(S) = {(1, 3), (9, 3), (1, 1), (1, 2), (2, 1), (5, 2), (6, 2)}

A set of spanning vectors contains α1 = (6, 2), α2 = (1, 3). R is not seminormal

since ω = (3, 1) /∈ S, but 2ω, 3ω ∈ S.

The spanning set T is illustrated by the large dots below.


-

6

cs ss s s s ss s s s s ss s sq q q q q q q qq q q q q qq q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q q qq q q q q q q q

��

��

��

��

(0,0)s α1

α2

S

Box is the parallelogram in the graph above. Obviously T is not in Box.

4.2. The Coincidence of CM and SN of Simplicial Semigroup Rings

It is not difficult to draw the conclusion below with all the preparation before.

Theorem 4.2.1: Let S ⊆ Nn be a simplicial affine semigroup with

G(S)∩◦

C(S)⊆ S, and rank(S) = m 6 n. Then R = k[S] Cohen-Macaulay implies

R seminormal.

Proof: Recall the notation in chapter 3: A =

α1

· · ·αm

, and Zm = Zn ∩ V,

where V denotes the linear subspace spanned by the spanning vectors of the cone

C(S) in Qn. Let h denote the index of the group G(S) in Zm.

Since R is Cohen-Macaulay, by Theorem 3.1.12, the spanning set T of S contains

det(A)/h elements. Then T must be contained in the Box according to Lemma 3.1.5.

Hence S is seminormal by Theorem 4.1.1. �

To explore the Cohen-Macaulay property of a simplicial seminormal affine semi-

group ring, we need Goto, Suzuki and Watanabe’s method.


4.2.1. A Mysterious Extension of Affine Semigroups. In 1976, Goto,

Suzuki and Watanabe [16] defined the following special extension of an affine semi-

group S to indicate the Cohen-Macaulay property of the corresponding affine semi-

group ring k[S].

Let S ⊆ Nn be an affine semigroup (not necessarily to be simplicial) with rank

r. Suppose spanning vectors of the cone C(S) are α1, · · · , αm. Define

Si = {ω ∈ G(S) |ω + β ∈ S for some β ∈ Fi ∩ S}.

where Fi is the i-th (r − 1)-dimensional facet of the cone C(S). Then an extension

of S is defined to be S ′ = ∩mi=1Si.

Example 4.2.2: Let S = {(x, y) ∈ N2 |x + y ≡ 0 mod 3, x 6= 1}. Then the

Hilbert basis of S is Hilb(S) = {(3, 0), (2, 1), (0, 3)}. Obviously the group generated

by S is G(S) = {(x, y) ∈ Z2 |x+ y ≡ 0 mod 3}, and the cone C(S) = Q2+.

The semigroup S is illustrated by the big solid dots in the following graph.

-

6

a a a a a a a a aa a a a a a a a aa a a a a a a a aa a a a a a a a aa a a a a a a a aa a a a a a a a aa a a a a a a a aa a a a a a a a a

t t tt t tt tt t tt t tt tt t tt t t

· · ·

...

(0,0) α1

α2

F2

F1

S

The facets of the cone are F1 = {(x, y) ∈ Q2+ |x = 0}, and F2 = {(x, y) ∈

Q2+ | y = 0}, i.e., F1, and F2 are non-negative y and x axes respectively.

By definition,

S1 = {(x, y) ∈ G(S) |x > 0, x 6= 1}


and

S2 = {(x, y) ∈ G(S) | y > 0}

Then S ′ = S1 ∩ S2 = S. �

In fact the equality between S and S ′ in above example is not a coincidence.

Goto, Suzuki and Watanabe gave the following criterion characterizing Cohen-

Macaulayness of simplicial semigroup rings.

Theorem 4.2.3: (Goto, Suzuki and Watanabe [16], Theorem 1) Let S ⊆ Nn

be a simplicial semigroup. Suppose spanning vectors of C(S) are α1, · · · , αm, and

rank(S) = m 6 n. Then following conditions are equivalent.

(1) R = k[S] is Cohen-Macaulay.

(2) For ω ∈ G(S), ω+αi ∈ S and ω+αj ∈ S, for some i and j (1 6 i < j 6 m),

implies ω ∈ S.

(3)⋂mi=1(αi + S) ⊂ (

∑mi=1 αi) + S.

(4) S = S ′. �

Example 4.2.4: (Example 4.2.2) The affine semigroup ring R = k[x3, x2y, y3]

in Example 4.2.2 is Cohen-Macaulay by above theorem. �

Following conditions hold for the extension S ′ of a not necessarily simplicial

affine semigroup S ⊆ Nn.

Fact 1. S ′ = S ′ ∩ C(S), i.e., S ′ ⊂ C(S). This is because S ′ = ∩mi=1Si, while

every Si contains only points which are in the non-negative side (or S side) of the

hyper-plane generated by Fi.

Fact 2. S ′ is simplicial if and only if S is simplicial.

Fact 3. (S ′)′ = S ′. By Theorem 4.2.3, k[S ′] is Cohen-Macaulay when S is

simplicial.


Fact 4. For an arbitrary affine semigroup S, if S is seminormal, S ′ is seminor-

mal.

To understand this, we need the following results.

Lemma 4.2.5: (Ewald [11], Chapter II, Theorem 1.11) Let C be an m-

dimensional polyhedral cone. Fix j, l ∈ Z+ with j 6 l < m. Then every j-

dimensional face F is an intersection of l-dimensional faces of C. In particular,

F is an intersection of codimension 1 facets of C.

Let C(S) be the cone generated by S with m facets F1, · · · , Fm. Define the i-th

hyper-plane of the cone, which contains Fi, to be Hi = {u ∈ Qn | fi(u) = 0}, where

fi is a linear polynomial fi : Qn → Q, i ∈ I such that C(S) = {u ∈ Qn | fi(u) >

0, for all i}. Then for any i ∈ I = [1,m], Fi = Hi ∩ C(S).

Proposition 4.2.6: Let S be a seminormal affine semigroup (not necessarily

simplicial). Then

Sp = G(S)fp>0 ∪G(S ∩ Fp)

where G(S)fp>0 denotes the set of elements ω in G(S) such that fp(ω) > 0, i.e.,

G(S)fp>0 is in the positive side of the hyperplane Hp.

Proof: “⊇”

• For each ω ∈ G(S)fp>0, we can always pick a β ∈ S∩◦Fp with fp(β) = 0,

and fi(β) > 0, i 6= p, such that fi(ω + β) > 0 for all i. This implies that

ω + β ∈◦

C(S). But ω + β ∈ G(S), so ω + β ∈ G(S)∩◦

C(S)⊂ S since S is

seminormal. Therefore ω ∈ Sp.• If ω ∈ G(S ∩ Fp), clearly fp(ω) = 0. Similarly pick β ∈ S∩

◦Fp satisfying

fp(ω+β) = 0, and fi(ω+β) > 0 for all i 6= p. Then ω+β ∈ G(S∩Fp)∩◦Fp.

Hence ω + β ∈ S by Theorem 2.4.8. So ω ∈ Sp.

“⊆” Suppose there is a ω ∈ Sp, but ω /∈ G(S)fp>0 ∪G(S ∩ Fp).


Observe that ω ∈ G(S)fp>0 = {γ ∈ G(S) | fp(γ) > 0}. This is because if not,

i.e., fp(ω) < 0, it would be impossible to get ω + β ∈ S for all β ∈ S ∩ Fp since

fp(ω + β) is always less than zero, so ω /∈ Sp which is contrary to the hypothesis.

Then since ω /∈ G(S)fp>0 ∪ G(S ∩ Fp), we have ω ∈ (G(S) ∩ Hp) \ G(S ∩ Fp).By definition of Sp, ω + β ∈ S for some β ∈ S ∩ Fp ⊆ G(S ∩ Fp). Notice that

ω + β ∈ S ∩ Fp ⊂ G(S ∩ Fp). So ω ∈ G(S ∩ Fp). A contradiction. �

We now describe S ′.

Proposition 4.2.7: Let S ⊆ Nn be a seminormal affine semigroup (not neces-

sarily simplicial). Then

S ′ = [G(S)∩◦

C(S)] ∪

(⋃F

[(⋂j∈J

G(S ∩ Fj)

)∩

◦F

]),

where F ranges over all proper faces of the cone C(S) and F = ∩j∈JFj, where J ⊂ I

is the set of indexes of all facets containing the face F .

Proof: “⊇”: It holds by the following fact.

For each Si, Proposition 4.2.6 tells us

G(S)∩◦

C(S)⊂ Si.

For each i, if the hyperplane Hi does not contain the face F , then the interior of

F , i.e.,◦F , lies strictly in the positive side of Hi. So[⋂

j∈J

G(S ∩ Fj)

]∩

◦F⊂ Si.

This is also true when Hi contains F , and in this case i ∈ J . Then[⋂j∈J

G(S ∩ Fj)

]∩

◦F⊂ G(S ∩ Fi)∩

◦F⊂ Si.

“⊆”: To prove this, we need to consider the following cases.

• Suppose ω ∈ S ′∩◦

C(S). Since S ′ ⊂ G(S), then ω ∈ G(S)∩◦

C(S).


• Let ω ∈ S ′∩◦F , F = ∩j∈JFj. Then ω ∈ Sj∩

◦F⊂ Sj ∩C(S) for all j ∈ J . By

Lemma 4.2.6, we have ω ∈ G(S ∩ Fj), j ∈ J . Hence ω ∈⋂j∈J G(S ∩ Fj).

So ω ∈[⋂

j∈J G(S ∩ Fj)]∩

◦F . �

This Proposition tells us that given G(S), S ′ is determined by the group gener-

ated by the lattice points of S on the codimension 1 facets and by G(S) itself if S

is seminormal.

Example 4.2.8: Let S = N3 \ {(0, 0, c) | c ≡ 1 mod 2, c ∈ N}. Then the

Hilbert basis of S is

Hilb(S) = {(1, 0, 0), (0, 1, 0), (0, 0, 2), (1, 0, 1), (0, 1, 1)}.

S is a simplicial affine semigroup. We take spanning vectors α1 = (1, 0, 0), α2 =

(0, 1, 0), α3 = (0, 0, 2). It is easy to see that S is seminormal by Theorem 2.4.8.

The semigroup S is illustrated below.

-

6

r r r r r rb r r r r rr r r r r rb r r r r rr r r r r r

(0,0,0)α2

sα3 s

S

F1F2

F3

��

r r r r r rr r r r r rr r r r r rr r r r r r

α1 = (1, 0, 0) s��

Clearly G(S) = Z3, and G(S ∩Fi) ∼= Z2 for i = 1, 2, 3. Therefore S ′ = N3 by the

above proposition. �


The following theorem holds immediately by applying Theorem 2.4.8 and Propo-

sition 4.2.7.

Proposition 4.2.9: Let S ⊆ Nn be an affine semigroup (not necessarily sim-

plicial). If S is seminormal, then S ′ is seminormal. �

Fact 5. When S ′ is seminormal, S ′ is not the seminormalization of S in general.

If it were, for every seminormal simplicial affine semigroup S, we would have that

S = S ′. So all seminormal simplicial semigroups would be Cohen-Macaulay, which

is not true.

Fact 6. S ′ is not normal in general. For instance, the semigroup in Example

4.2.2.

Fact 7. For an arbitrary affine semigroup S, it is not necessary that +S ⊆ S ′,

where +S denotes the seminormalization of S. An easy example can be found

in Example 4.2.2 in which S ′ = S and S is not seminormal. By Theorem 2.4.8,

+S = G(S) ∩ N2 = S, where S denotes the normalization of S. We have S ′ ⊂ +S.

If S is seminormal, then S = +S ⊆ S ′ ⊆ S.

Fact 8. If S is simplicial, then S ′ is a minimal Cohen-Macaulay extension of S.

Theorem 4.2.10: Let S be a simplicial affine semigroup. Then k[S ′] is the

minimal Cohen-Macaulay semigroup ring which contains k[S].

Proof: Suppose there is an extension, denoted by S∗, of S such that k[S∗] is

Cohen-Macaulay and S ⊆ S∗ ( S ′.

It is easy to show that S ⊆ S∗ implies S ′ ⊆ (S∗)′. Then we have S∗ ( S ′ ⊆ (S∗)′.

But (S∗)′ = (S∗) since k[S∗] is Cohen-Macaulay. Hence S∗ = S ′. A contradiction.

�


4.2.2. The Cohen-Macaulay Property of Seminormal Simplicial Affine

Semigroup Rings. Clearly affine semigroups with rank 1 and 2 are always sim-

plicial. In fact, a seminormal affine semigroup with rank 1 is normal, hence is

Cohen-Macaulay.

According to Theorem 4.2.3 and Proposition 4.2.7, when S is a simplicial semi-

normal semigroup, the Cohen-Macaulay property of the semigroup ring R = k[S] is

determined by the codimension 1 faces of S and the group generated by S.

Proposition 4.2.11: Let S ⊆ Nn be a seminormal simplicial affine semigroup

with rank 2. Then R = k[S] is a Cohen-Macaulay ring.

Proof: Since rank(S) = 2, by Lemma 4.2.7,

S ′ = [G(S)∩◦

C(S)] ∪

[2⋃i=1

G(S ∩ Fi)∩◦Fi

]∪ {0}.

However, this is nothing but +S by Theorem 2.4.8. So S ′ = +S = S. Hence R = k[S]

is Cohen-Macaulay by Theorem 4.2.3. �

With this nice theorem, the Cohen-Macaulay property and seminormality coin-

cide for such a simplicial semigroup S that S ⊇ G(S)∩◦

C(S) if rank(S) 6 2.

Theorem 4.2.12: Let S be a simplicial semigroup containing all elements of

G(S)∩◦

C(S) with rank(S) 6 2. Then R is Cohen-Macaulay if and only if R is

seminormal. �

Remark: This result is not true in general when m > 3.

Example 4.2.13: (Example 4.2.8) R = k[x, y, z2, xz, yz] is seminormal but not

Cohen-Macaulay.

Proof: By using the computer program TotalProgram introduced in Appendix

B, The spanning set T of S contains following elements:

T = {(0, 0, 0), (0, 1, 1), (1, 0, 1)}


Now the cardinality of the spanning set T is 3, but the determinant of A =

α1

α2

α3

is 2. So R = k[S] is not a Cohen-Macaulay ring by Theorem 3.1.12. �

Theorem 4.2.14: Let S ⊆ Nn be a seminormal affine semigroup. Then the

following two conditions are equivalent,

(1) S = S ′.

(2) S∩◦F=

⋂j∈J G(S ∩ Fj)∩

◦F , for all proper faces F = ∩j∈JFj of the cone

C(S), where J ⊂ I is the set of indices of all facets containing the face F .

�

Corollary 4.2.15: Let S ⊆ Nn be a seminormal simplicial affine semigroup.

Then R = k[S] is Cohen-Macaulay if and only if S∩◦F=

⋂j∈J G(S ∩ Fj)∩

◦F for

every proper face F = ∩i∈JFi of the cone C(S), where J ⊂ I is the set of indices of

all facets containing the face F . �

Example 4.2.16: (Example 4.2.8)R = k[x, y, z2, xz, yz] is not Cohen-Macaulay.

Proof: This is a new approach to the Cohen-Macaulayness of the semigroup S

in Example 4.2.8.

As we saw before, S = N3 \{(0, 0, c) | c ≡ 1 mod 2, c ∈ N}, and S is a simplicial

semigroup. Let Fxy, Fxz, Fyz denote the xy, xz, xy coordinate planes, and Fx, Fy, Fz

denote the x, y, z axes respectively.

It is easy to show that G(S ∩ Fxz) = G(S ∩ Fyz) = Z2, and Fz = Fxz ∩ Fyz.Then G(S ∩ Fxz) ∩ G(S ∩ Fyz)∩

◦Fz= N \ {0}. But S∩

◦Fz= 2N \ {0}. Hence

G(S ∩ Fxz) ∩ G(S ∩ Fyz)∩◦Fz! G(S ∩ Fz)∩

◦Fz. According to Corollary 4.2.15,

R = k[S] is not a Cohen-Macaulay ring. �


4.3. Seminormality of Affine Semigroups

In section 2, we have seen some interesting features of the mysterious extension

S ′ of S. In this section, we shall find that the coincidence between S ′ and S not only

characterizes the Cohen-Macaulay property of simplicial affine semigroup rings, but

also indicates the seminormality of arbitrary affine semigroups under a restricted

hypothesis.

Theorem 4.3.1: Let S ⊂ Nn be an arbitrary affine semigroup withG(S)∩◦

C(S)⊂S. rank (S) = r 6 n. Then S ′ = S implies S is seminormal.

Proof: Assume the semigroup S is not seminormal. Then the non-seminormal

points must be in some proper faces of S by the hypothesis. We may take ω ∈G(S)∩

◦F , where F is a proper face of the cone C(S), such that 2ω, 3ω ∈ S, but

ω /∈ S.

Suppose there are m codimension 1 facets Fi in the cone C(S). Same as before,

for each i ∈ I = [1,m], we have Fi = Hi ∩ C(S), where Hi = {u ∈ Qn | fi(u) = 0 },and fi : Qn → Q is a linear polynomial. Furthermore, C(S) = {u ∈ Qn | fi(u) >

0, for all i }.

Let J ⊂ I be the set of all the integers j such that ω ∈ Fj. Then ω ∈ Sj for

j ∈ J . This is because 2ω, 3ω ∈ Fj ∩ S implies dω ∈ Fj ∩ S for all d ∈ N, d > 1.

Hence ω+ dω = (d+ 1)ω ∈ S for some positive integers d > 1. Therefore ω ∈ Sj by

the definition of Sj.

Given an integer p ∈ I \ J , pick an element β ∈ S∩◦Fp so that fp(β) = 0

and fi(β) > 0 for all i 6= p. Let we consider ω + β. When i 6= p, i ∈ I, we have

fi(ω+β) = fi(ω)+fi(β) > 0. Well, for the integer p, since ω /∈ Fp, but ω ∈ C(S), so

fp(ω) > 0, which implies fp(ω + β) > 0. Then ω + β ∈ G(S)∩◦

C(S)⊂ S. Therefore

ω ∈ Sp.


Now we have ω ∈ Si for all i ∈ I. Hence ω ∈ ∩mi=1Si = S ′ = S which is a

contradiction. �

Remark: The hypothesis G(S)∩◦

C(S)⊂ S is necessary. For instance, in Exam-

ple 4.2.2, we have shown S = S ′, but S is not seminormal by Theorem 2.4.8.

4.4. The Coincidence between SN and CM of Arbitrary Semigroup

Rings

Let us begin from an introduction of the Cousin Complex. This is to explain the

equivalence of (1) and (4) in Theorem 4.2.3.

Let R be a Zn-graded commutative Noetherian ring. Recall that a Cousin Com-

plex (Sharp [35]) of an Zn-graded R-module M is a complex

C(M) : 0d−2

// M−1 = Md−1

// M0 // · · · // Mn−2 dn−2// Mn−1

It is constructed inductively as follows:

DefineM i to be the direct sum of the Zn-homogeneous localizations of coker(di−2)

at all graded prime ideals ℘ of R with ht(℘) = i, i.e.,

M i =⊕

℘∈Supp(M),htM (℘)=i

(coker di−2)℘.

The homomorphisms di−1 is defined to be the composition of the canonical epimor-

phism M i−1 → coker (di−2) and φ : coker (di−2) →M i for which, if x ∈ coker (di−2),

the component of φ(x) is ⊕x1.

Sharp showed in his paper [35] that Cohen-Macaulay rings can be characterized

in terms of the Cousin Complex. More precisely,

Theorem 4.4.1: (Sharp [35]) The ring A is Cohen-Macaulay if and only if its

Cousin Complex, C(A), is exact.


This result is not for graded rings, but it can be extended to the Zn-graded case

as found in Goto and Watanabe’s paper “On graded rings, I” [17], and especially

“On graded rings, II: Zn-graded rings”[18].

For an arbitrary affine semigroup S, we know that the corresponding semigroup

ring R = k[S] is Zn-graded and Noetherian. We calculate the Cousin Complex

of the semigroup ring R as follows; M0 = R(0), and M1 = ⊕ht (℘)=1(R(0)/R)℘ =

⊕ht (℘)=1(R(0)/R℘). Trung and Hoa proved that ker d0 = k[S ′] (Lemma 2.1, [46]).

Then by exactness of Cousin Complex C(R),

Theorem 4.4.2: (Trung and Hoa [46]) S = S ′ if R = k[S] is a Cohen-Macaulay

ring.

To get the sufficient condition of the Cohen-Macaulay property, one has to add

some topological condition on the convex cone C(S) besides S ′ = S.

We need some more notation adopted from Trung and Hoa [46]. Let C(S) be

the cone generated by an affine semigroup S with m facets F1, · · · , Fm. For every

subset J of I = [1,m], define

GJ =⋂i/∈J

Si \⋃j∈J

Sj.

Let πJ be the simplicial complex of nonempty subsets L of J with the property

∩i∈LS ∩ Fi 6= (0). We call πJ acyclic if the reduced homology group Hq(πJ ; k)

vanishes for all q > 0.

Theorem 4.4.3: (Trung and Hoa [46]) Let S be an arbitrary affine semigroup

with rank(S) = r. Let C(S) be the cone spanned by m spanning vectors α1, · · · , αm.

Then K[S] is a Cohen-Macaulay ring if and only if following conditions (1) and (2),

or equivalently, (1) and (3) are satisfied:

(1) S ′ = S.

(2) For every nonempty proper subset J of I = [1,m], GJ = ∅ or πJ is acyclic.


(3) For every nonempty subset J of I with J /∈ πI and #J 6 m− 2, GJ = ∅ or

Hq(πJ ; k) = 0 for all q < r − 2.

�

The corollary below is a consequence of Theorem 4.3.1 by the above theorem. It

generalizes Theorem 4.2.1 to the non-simplicial case.

Corollary 4.4.4: Let S ⊆ Nn be an arbitrary affine semigroup withG(S)∩◦

C(S)⊂S. Then R = k[S] Cohen-Macaulay implies R seminormal. �

Next we shall consider the Cohen-Macaulay property of seminormal semigroup

rings. Let us begin from the counter example to the result of Goto and Watanabe

given by Trung and Hoa [46].


Hilb(S) = {(3, 0, 0), (2, 1, 0), (0, 3, 0), (3, 0, 1), (2, 1, 1), (0, 3, 1)}.

Then the corresponding semigroup ring is R = k[x3, x2y, y3, x3z, x2yz, y3z] = A/I

where A = k[t1, · · · , t6]. Trung and Hoa proved that S = S ′, but R is not Cohen-

Macaulay. This can be seen by using the graded Auslander-Buchsbaum Theorem

(Theorem A.2.1) and Macaulay2. The code is cited below.

i1 : k=QQ

o1 = QQ

o1 : Ring

-- the class of all rational numbers

i3 : B=k[x,y,z]

o3 = B

o3 : PolynomialRing

i4 : A=k[t_1..t_6, Degrees=>{3,3,3,4,4,4}]

o4 = A

o4 : PolynomialRing

i5 : f=map(B,A,{x^3,x^2*y,y^3,x^3*z,x^2*y*z,y^3*z})

3 2 3 3 2 3


o5 = map(B,A,{x , x y, y , x z, x y*z, y z})

o5 : RingMap B <--- A

i6 : I=ker f

o6 : Ideal of A

i7 : R=A^1/module(I)

o7 = cokernel | t_3t_5-t_2t_6 t_3t_4-t_1t_6 t_2t_4-t_1t_5 t_2^3-t_1^2t_3

t_2^2t_5-t_1^2t_6 t_2t_5^2-t_1t_4t_6 t_5^3-t_4^2t_6 |

o7 : A-module, quotient of A

i8 : res R

1 7 11 6 1

o8 = A <-- A <-- A <-- A <-- A <-- 0

0 1 2 3 4 5

o8 : ChainComplex

From Line i8, we know the projective dimension of R is 4. So by the graded

Auslander-Bachsbaum Theorem (Theorem A.2.1),

depth(R) = depth(A)− proj dim(R) = 6− 4 = 2

which is less than the Krull-dimension 3 of R. Hence R is not Cohen-Macaulay.

Note that the element (1, 2, 0) ∈ G(S)∩◦

C(S), but (1, 2, 0) /∈ S. Then according

to Theorem 2.4.8, S is not seminormal. �

Inspired by this example, we may give the following conjecture.

Conjecture: Let S be an arbitrary seminormal affine semigroup. Then S = S ′

implies that the semigroup ring R = k[S] is Cohen-Macaulay.

As a preparation, we need the following lemma.

Lemma 4.4.6: Let S be an arbitrary affine seminormal semigroup. Let J be a

nonempty subset of [1,m] such that⋂j∈J(Fj

⋂S) = (0), i.e., J /∈ πI . Assume that

πJ is not acyclic. If there is an α ∈ GJ , α 6= 0, then the following two conditions

hold.

(1) α /∈ C(S).


(2) α ∈ Hj for some j ∈ J . �

Proof: Clearly α should be in the non-negative side of Hi for i /∈ J and in the

non-positive side of Hj for j ∈ J by S seminormal and Lemma 4.2.6.

(1) Suppose α ∈ C(S). Then α ∈⋂j∈J Fj since α should be in the non-positive

side of Hj for all j ∈ J . Hence F :=⋂j∈J Fj is a proper face of the cone

C(S). Note that C(S) is the smallest cone generated by S. This implies

that S⋂F =

⋂j∈J(S

⋂Fj) 6= (0) which contradicts the hypothesis J /∈ πI .

Hence the condition (1) follows.

(2) Take the normalization S of S. From now on, we will top bar all notations

relating to S to indicate the corresponding ones of S. Obviously πJ = πJ ,

since S and S have the same topological features.

It is well known that k[S] is Cohen-Macaulay. Then by Theorem 4.4.3,

when πJ is not acyclic,

GJ =⋂i/∈J

Si \⋃j∈J

Sj = ∅,

where Si = G(S)fi>0 ⊇ Si. This implies that⋂i/∈J

Si \⋃j∈J

Sj = ∅.

Note that⋃j∈J Sj ⊂

⋃j∈J Sj. Then if there is an α ∈ GJ , this α must be

in⋃j∈J Sj \

⋃j∈J Sj which implies that α ∈ Hp

⋂G(S) for some p ∈ J .

�

Theorem 4.4.7: Let S be an arbitrary affine semigroup with rank r = 3. Then

S seminormal and S = S ′ imply the semigroup ring R = k[S] is Cohen-Macaulay.

Proof: By Theorem 4.4.3, it is enough to show that for all nonempty subsets

J ⊆ I with J /∈ πI and #J 6 m− 2, GJ = ∅ when H0(πJ ; k) 6= 0.

Suppose not, i.e., there are J and α ∈ GJ while H0(πJ ; k) 6= 0. Clearly α 6= (0).


Since H0(πJ ; k) 6= 0, without loss of generality, we may assume that there are

two subcomplexes πJ1 and πJ2 of πJ such that πJ = πJ1 ∪ πJ2 and πJ1 ∩ πJ2 = ∅,where J = J1 ∪ J2, J1 ∩ J2 = ∅. It follows that there are two groups of facets FJ1

and FJ2 which are not adjacent.

According to Lemma 4.4.6, we may assume α ∈ Hj1 , where j1 ∈ J1. Now

we extend all facets of the cone C(S), and look at the intersection lines of these

hyperplanes with Hj1 . Obviously these lines intersect at the origin. The graphs are

illustrated as follows.

@@

BBBBBB

��

��

��

��

@@@@@@@@@

BBBBBBBB

��

��r r r r r r r r

Fj1H1

F1

H2

F2

Hj1

C(S)

Figure 1. The cross section of the cone C(S)

Fj1H1 ∩Hj1 H2 ∩Hj1

Hi ∩Hj1

Hj1

Figure 2. The hyperplane Hj1

The shaded region represents the facet Fj1 of the cone. Clearly the positive side of

the line Hi ∩ Hj1 , for any i 6= j1, on the hyperplane Hj1 is the side toward to the

facet Fj1 .

Let F1 and F2 be two facets of the cone C(S) whose indices are not in J , but

are adjacent to FJ1 . Note that the line H1 ∩Hj1 or H2 ∩Hj2 can be the boundary


of Fj1 . In Figure 1, when we consider the distance of the intersection point of the

hyperplane Hi and Hj1 to the facet Fj1 on the cross sections of C(S) and Hj1 , it is

obviously that the closer the facet Fi (corresponding to the hyperplane Hi) is to the

facet Fj1 , the smaller the distance is. Referring to Figure 2, this fact is reflected by

the position of lines Hi ∩Hj1 for all i ∈ I, i 6= j1. More precisely, on the hyperplane

Hj1 , the line Hj ∩Hj1 cuts H+1 ∩H+

2 ∩Hj1 for all j ∈ J1 \ j1, and if i /∈ {1, 2} ∪ J1,

the line Hi ∩Hj1 is in[H<0

1 ∪H<02 ∪ (0)

]∩Hj1 , i.e.,

H−i ∩Hj1 ⊂

[H<0

1 ∪H<02 ∪ (0)

]∩Hj1

(From now on, notations H+i and H>0

i (or H−i and H<0

i ) represent the non-negative

side and positive side (or non-positive side and negative side) of the hyperplane Hi

respectively.)

Since α ∈ GJ =⋂i/∈J Si \

⋃j∈J Sj and S is seminormal, α must be in the

intersection of the non-negative sides of hyperplanes H1 and H2 by Lemma 4.2.6,

i.e.,

α ∈[H+

1

⋂H+

2

]⋂Hj1

On the other hand, α ∈ GJ and S seminormal imply α is in the intersection of

the non-positive sides of hyperplanes Hj with i ∈ J2, i.e.,

α ∈

[⋂i∈J2

H−i

]⋂Hj1 .

In fact, if i ∈ J2, then i /∈ {1, 2} ∪ J1. Therefore, by the previous discussion,

H−i ∩Hj1 ⊂

[H<0

1 ∪H<02 ∪ (0)

]∩Hj1 . It follows that[⋂

i∈J2

H−i

]⋂Hj1 ⊂

[H<0

1

⋃H<0

2

⋃(0)]⋂

Hj1 .

Hence α ∈[H<0

1 ∪H<02 ∪ (0)

]∩Hj1 . A contradiction.

CHAPTER 5

The Canonical Module

In this chapter, we shall investigate the canonical module of Cohen-Macaulay

simplicial affine semigroup rings. We were guided by the book of Bruns and Herzog

[7] and the paper of Goto, Suzuki and Watanabe [16] in the first section in which

basic concepts and properties are introduced. For a simplicial affine semigroup ring

R = k[S], an ideal KS of S was given by Goto, Suzuki and Watanabe, and they

proved that theR-module k[KS] is the canonical module ofR ifR is Cohen-Macaulay

([16], Theorem3.8). In section 2, we shall show that the geometrical appearance of

KS can be used to characterize seminormality of simplicial semigroup rings. The

structure of KS is clarified when S is a seminormal affine semigroup.

5.1. The Canonical Module

Let S ⊆ Nn be an affine semigroup. Let G(S) denote the group generated by

the semigroup S in Zn. Let U, V be two subsets of G(S). We define

U + V = {u+ v |u ∈ U and v ∈ V }

−U = {−u |u ∈ U}

ω + U = {ω + u |u ∈ U}, for all ω ∈ S.

Define XU to be the vector subspace of k[S] whose k-basis consists the set of

elements xu with u ∈ U .

73


In an affine semigroup S, we may define ideals: A subset U of the semigroup S

is called an ideal of S if S + U ⊂ U . An ideal U of S is said to be finitely generated

if there is a finite number of elements u1, · · · , un ∈ S such that U = ∪ni=1(ui + S).

The following proposition shows the correspondence between the ideal theory of

an affine semigroup S and that of the semigroup ring k[S].

Proposition 5.1.1: (Bruns and Herzog [7] Proposition 6.1.1) Let S be an

affine semigroup, and U a subset of S. Then U is an ideal of S if and only if XU is

an ideal of k[S]. �

Remark: (Goto, Suzuki and Watanabe [16])

(1) If U is an ideal of an affine semigroup S, XU is a Zn-graded R module.

(2) If U is finitely generated, XU is finitely generated.

Let R be a ring and M an R-module. A complex

I.: 0 → I0 → I1 → I2 → · · ·

with injective modules I i is an injective resolution of M if H0(I.) ∼= M and H i(I.) =

0 for i > 0. The injective dimension of M , denoted inj dim M , is the smallest integer

n for which there exists an injective resolution I · of M with Im = 0 for m > n. If

there is no such n, the injective dimension of M is infinite.

Definition 5.1.2: Let R be a Cohen-Macaulay local ring. A maximal Cohen-

Macaulay module C of type 1 and of finite injective dimension is called a canonical

module of R.

If R is a arbitrary Cohen-Macaulay ring. A finitely generated R-module M is a

canonical module of R if Mm is a canonical module of Rm for all maximal ideals m

of R. �


Recall that a positively Z-graded ring is a ring R together with a decomposition

R =⊕

i∈NRi such that RiRj ⊂ Ri+j for all i, j ∈ N. A positively graded ring with

only one homogeneous maximal ideal m is denoted by (R,m).

For a graded ring (R,m), the canonical module over R is defined as follows.

Definition 5.1.3: Let (R,m) be a graded ring of dimension d. A finitely

generated R-module C is a canonical module of R if there exist homogeneous iso-

morphisms

ExtiR(R/m, C) ∼= { 0 for i 6= dR/m for i = d

�

This definition is consistent with Definition 5.1.2, and the proof can be found in

[7] by Bruns and Herzog ([7] Definition 3.6.8 and Proposition 3.6.9).

Recall that the type of a finitely generated non-zero graded R-module of depth

d is defined to be the number dimR/mExtdR(R/m,M). Then Definition 5.1.3 is equiv-

alent to the following definition which can be found in Herzog and Kunz’s lecture

notes [23], or Goto, Suzuki and Watanabe’s paper ([16] page 7).

Definition 5.1.4: Let R be a graded Cohen-Macaulay ring of dimension d.

Then the canonical module C of R is defined by the following conditions

(1) C is a finitely generated Cohen-Macaulay R-module of dimension d, and

(2) the type of C is 1. �

5.2. The Canonical Module of a Cohen-Macaulay Seminormal

Semigroup Ring

Let S be a simplicial affine semigroup. Let the spanning vectors of C(S) be

α1, · · · , αm. In [16] by Goto, Suzuki and Watanabe, a special finitely generated


ideal KS of S was constructed as KS = − ∩mi=1 Ci, where Ci := G(S) \ Si, and the

following theorem holds.

Theorem 5.2.1: (Goto, Suzuki and Watanabe [16], Theorem 3.8)

Let R = k[S] be a Cohen-Macaulay simplicial affine semigroup ring. Then k[KS]

is the canonical module of R. �

When S is a normal affine semigroup, S = G(S)∩C(S). By definition, for each

i ∈ I, Si contains all lattice points of the group G(S) on the non-negative side of

the hyperplane Hi = {u ∈ Qn | fi(u) = 0 }. It follows that Ci = G(S)fi<0, where

G(S)fi<0 denotes the set of lattice points of the group G(S) on the negative side of

the hyperplane Hi. Hence KS = − ∩i Ci = G(S)∩◦

C(S).

What does KS look like if S is not normal? Here are some easy examples.


Hilb(S) = {(2, 0), (0, 3), (1, 1), (2, 1), (1, 2), (1, 3)}

Then the cone C(S) = Q2+. The codimension 1 facets of the cone C(S), denoted by

F1 and F2, are the positive y and x axes respectively. Clearly S is seminormal.

The semigroup S contains all interior lattice points in the cone C(S) and the

non-negative integer multiples of spanning vectors α1 = (2, 0) and α2 = (0, 3). By

Theorem 3.2.1, the corresponding semigroup ring k[S] is Cohen-Macaulay.

Now we calculate KS.

It is easy to see that

S1 = {(x, y) ∈ Z2 |x > 0 or if x = 0, y ≡ 0 mod 3};S2 = {(x, y) ∈ Z2 | y > 0 or if y = 0, x ≡ 0 mod 2}.

Then

C1 = {(x, y) ∈ Z2 |x < 0 or if x = 0, y ≡ 1, 2 mod 3};C2 = {(x, y) ∈ Z2 | y < 0 or if y = 0, x ≡ 1 mod 2}.


So

C1 ∩ C2 = {(x, y) ∈ Z2 |x < 0 and y < 0}∪{(x, y) ∈ Z2 |x = 0, y < 0, y ≡ 1, 2 mod 3}∪{(x, y) ∈ Z2 | y = 0, x < 0, x ≡ 1 mod 2}.

Therefore KS = −(C1 ∩ C2) contains all integer points in the interior of C(S), all

integer points on the positive x axis except every second point, and all integer points

on the positive y axis except every third point. Geometrically KS is illustrated by

the solid dots in the following diagram.

-

6

s s s ss s s s s s s ss s s s s s s ss s s s s s ss s s s s s s ss s s s s s s ss s s s s s s

c c c c

c

cq q q

qqq

x

y

KS

�

The above example tells us thatKS ⊂ C(S). However, the idealKS is not necessarily

in C(S).

Example 5.2.3: (Example 4.2.4) Let S = {(x, y) ∈ N2 |x+ y ≡ 0 mod 3, x 6=1}. The spanning vectors of C(S) are α1 = (3, 0), α2 = (0, 3). This semigroup is

not seminormal since (1, 2) /∈ S, but 2(1, 2) and 3(1, 2) ∈ S. The corresponding

semigroup ring R is Cohen-Macaulay.

It is easy to calculate S1 and S2.

S1 = {(x, y) ∈ Z2 |x+ y ≡ 0 mod 3, x > 0, x 6= 1};S2 = {(x, y) ∈ Z2 |x+ y ≡ 0 mod 3, y > 0}.

It follows that

C1 = {(x, y) ∈ Z2 |x+ y ≡ 0 mod 3, x < 0, or x = 1};C2 = {(x, y) ∈ Z2 |x+ y ≡ 0 mod 3, y < 0}.


Hence

KS = −(C1 ∩ C2) = {(x, y) ∈ Z2 |x+ y ≡ 0 mod 3, x > 0, y > 0}∪{(−1, y) ∈ Z2 | y ≡ 1 mod 3, y > 0}

which is represented by the solid dots below.

-

6

b b b b b b b b b b b bb b b b b b b b b b b bb b b b b b b b b b b bb b b b b b b b b b b bb b b b b b b b b b b bb b b b b b b b b b b bb b b b b b b b b b b bb b b b b b b b b b b bb b b b b b b b b b b b

s s s ss s ss s ss s s ss s ss s ss s s ss s sp p p

ppp

x

y

KS

Note that KS * C(S) in this example. �

We may ask the following question;

Question: Is this phenomenon related to the seminormality of the affine semi-

group ring?

The answer is yes. Precisely, if S is seminormal, then KS ⊂ C(S).

Theorem 5.2.4: Let S be a affine semigroup (not necessarily simplicial). If S

is seminormal, then KS ⊆ G(S) ∩ C(S).

Proof: If S is seminormal, recall that Si = G(S)fi>0 ∪ G(S ∩ Fi) for all i ∈ I by

Proposition 4.2.6. Geometrically, Si contains all lattice points of the group G(S) on

the positive side of the hyperplane Hi, and all lattice points of group G(S∩Fi) on the

hyperplane. Then Ci = G(S)\Si should be on the non-positive side of the hyperplane

Hi which implies −Ci is on the non-negative side of Hi for all i. When we take the

intersection ∩mi=1(−Ci) = − ∩mi=1 Ci = KS, it follows that KS ⊆ C(S) ∩G(S). �


In fact, if the semigroup ring R = k[S] is Cohen-Macaulay, the canonical module

k[KS] of R can be used to characterize the seminormality of the simplicial affine

semigroup S.

Theorem 5.2.5: Let S ⊆ Nn be a simplicial affine semigroup, and the corre-

sponding semigroup ringR = k[S] a Cohen-Macaulay semigroup ring. Let α1, · · · , αmbe spanning vectors of the cone C(S). Then S is seminormal if and only if KS ⊆G(S) ∩ C(S). �

Proof: By Theorem 5.2.4, it is enough to prove that KS ⊆ G(S) ∩ C(S) implies S

is seminormal.

Assume that S is not seminormal. Then some non-seminormal points occur in

the interior of C(S). This is because if G(S)∩◦

C(S)⊂ S, and S is Cohen-Macaulay,

then S must be seminormal by Theorem 4.2.1.

Take ω ∈ G(S)∩◦

C(S) such that 2ω, 3ω ∈ S, but ω /∈ S. Notice that S = ∩mi=1Si

since R is Cohen-Macaulay. So ω /∈ ∩mi=1Si, which implies that there is an integer

p, 1 6 p 6 m, such that ω /∈ Sp. By the definition of Sp, we know ω + β /∈ S for

all β ∈ Fp ∩ S. Then ω − β /∈ Sp for all β ∈ Fp ∩ S. This can be seen as follows: If

ω − β ∈ Sp, then ω = (ω − β) + β ∈ Sp which contradicts to the assumption.

We may assume that ω =∑m

i=1 λiαi, λi ∈ Q>0, for all i ∈ I, since ω ∈◦

C(S).

Pick β =∑

i6=p µiαi ∈ Fp∩S with µi ∈ Q>0 for all i so that λi−µi < 0, i ∈ I, i 6= q.

We claim that ω − β /∈ Sq for all q ∈ I, q 6= p. This will be clear by the following

calculation, where βq =∑

i6=q γiαi ∈ Fq ∩ S, γi ∈ Q+.

ω − β + βq =∑i6=p,q

(λi − µi + γi)αi + (λq − µq)αq + (λp + γp)αp /∈ S

since λq − µq < 0.

Then ω−β ∈ ∩mi=1Ci, i.e., β−ω ∈ KS. Obviously β−ω =∑

i6=p(µi−λi)αi−λpαp /∈C(S). Therefore KS 6⊆ G(S) ∩ C(S). A contradiction. �


5.3. The Structure of KS

Theorem 5.3.1: Let S be an affine seminormal semigroup (not necessarily

simplicial). Then

KS = [G(S)∩◦

C(S)]⋃(⋃

F

[(G(S)∩

◦F ) \

⋃i∈J

G(S ∩ Fi)

])where F ranges over all proper faces of the cone C(S), F = ∩i∈JFi, and J is the set

of indices of all facets containing F .

Proof: “⊆” By Theorem 5.2.4, KS ⊆ G(S) ∩ C(S).

If ω ∈ KS∩◦

C(S), then ω ∈ G(S)∩◦

C(S).

If ω ∈ KS∩◦F , where F = ∩i∈JFi, and J is the set of indices of all facets

containing F , then −ω /∈ Si, for all i ∈ I. (This is true just because ω ∈ KS =

−∩i∈I Ci. It follows that −ω ∈ Ci = G(S) \Si, for all i.) Therefore −ω /∈ G(S ∩Fi)if i ∈ J by Proposition 4.2.6. Note that G(S ∩ Fi) is a group, so ω /∈ G(S ∩ Fi) for

all i ∈ J . This implies that ω ∈ (G(S)∩◦F ) \ [∪i∈JG(S ∩ Fi)].

“⊇” It follows by these facts below.

G(S)∩◦

C(S)⊂ −Ci for all i ∈ I.

For each proper face F = ∩i∈JFi of the cone C(S), let

D = (G(S)∩◦F ) \

⋃i∈J

G(S ∩ Fi).

Since Cp = G(S) \ Sp, and Sp = G(S)fp>0 ∪G(S ∩Fp), then −Cp ∩Hp = Cp ∩Hp =

[G(S) ∩Hp] \G(S ∩ Fp). Clearly for p ∈ J ,

G(S)∩◦F ⊂ G(S) ∩Hp⋃

i∈J G(S ∩ Fi) ⊃ G(S ∩ Fp)

HenceD = (G(S)∩◦F )\

⋃i∈J G(S∩Fi) ⊂ [G(S)∩Hp]\G(S∩Fp) = −Cp∩Hp ⊂ −Cp,

for p ∈ J.


If p /∈ J , D is on the positive side of HP , so D ⊂ −Cp. Hence D ⊂ − ∩i∈I Ci =

KS. �

Remark: We don’t know the meaning of KS when S is not simplicial, and

we find no reference for this. It would be an interesting question to explore in the

future.

CHAPTER 6

Summary and Future Work

The main problem treated in this thesis is the coincidence between the Cohen-

Macaulay property and seminormality of affine semigroup rings.

The conjecture in Chapter 4 has not been proved for the affine semigroup with

rank > 3. According to Trung and Hoa’s theorem (Theorem 4.4.3 in this thesis),

other than S = S ′, the Cohen-Macaulay property of a seminormal semigroup ring

is mainly determined by the combinatorial feature of the cone spanned by the cor-

responding semigroup which is the same as the one of the normalization of the

semigroup. Therefore if there was something bad happened in seminormal case,

then the same thing would happen in normal case which is not true. The conjecture

is anticipated to be true. I would like to continue my research on this topic in the

future.

The work on the canonical module in Chapter 5 is intended to be a starting

point for future work on the topics of the canonical module, dualizing complexes

and Gorenstein rings. There are two directions that can be possibly followed. One of

these is to continue the work on canonical modules of Cohen-Macaulay seminormal

affine semigroup rings. Another interesting topic is to explore the dualizing complex

for non-Cohen-Macaulay seminormal semigroup rings based on the structure of KS.

Hilbert functions of affine semigroup rings is also an interesting field. We know

that affine semigroup rings can be treated as multi-graded rings, and the degrees

of the generators of an affine semigroup ring are usually not 1. This fact causes

difficulties when we work on Hilbert functions of semigroup rings. One possible way

82

Chapter . Summary and Future Work 83

to unlock this is to use the tool of homogenization. Some work has been done by

Leslie Roberts and Patil ([30] by Roberts, [34] by Patil and Roberts). Generalizing

those ideas in these papers is also interested.

We know that Buchsbaum rings have many similar properties to those of Cohen-

Macaulay rings. Among other possible fields for future study, the connections of

the Buchsbaum property and seminormality of affine semigroup rings should draw

more attention. However many examples show that the Buchsbaum property and

seminormality are not coincident.

Example 6.0.2: R = k[x, y, z2, xz, yz] is seminormal but not Buchsbaum.

S = N3 \ {(0, 0, c) | c ≡ 1 mod 2, c ∈ N}. S is a simplicial semigroup with spanning

vectors α1 = (1, 0, 0), α2 = (0, 1, 0), α3 = (0, 0, 2). S is seminormal by the previous

example (Example 4.2.8). Take ω = (0, 0, 1) ∈ G(S). Clearly ω+ 2α1, ω+ 2α2 ∈ S.

But ω + α3 /∈ S. So R is not Buchsbaum by Theorem 2.3.11.

Example 6.0.3: S is a simplicial semigroup with Hilbert basis

Hilb(S) = {(1, 3), (9, 3), (1, 1), (1, 2), (2, 1), (5, 2), (6, 2)}

The spanning vectors are α1 = (1, 3), α2 = (9, 3). Obviously R is not seminormal

since ω = (3, 1) /∈ S, but 2ω, 3ω ∈ S. While R is Buchsbaum by Theorem 2.3.11.

This is undoubtedly a difficult problem.

APPENDIX A

Some Criteria for the Cohen-Macaulay Property

This appendix intends to present several practical methods to determine whether

or not an affine semigroup ring is Cohen-Macaulay. The computer programs used

in this appendix are Macaulay 2 and TotalProgram introduced in appendix B.

Macaulay 2 can be obtained free from Macaulay 2 home page:

http://www.math.uiuc.edu/Macaulay2

and the program TotalProgram works under the Mathematica environment.

A.1. Using the Definition of Cohen-Macaulay Rings

According to the definition of Cohen-Macaulay Rings, we may use a computer

program to decided if a positive graded ring (R,m) is Cohen-Macaulay, in other

words, find the depth of R, i.e., depth(R) = min{i; ExtiR(k,R) 6= 0} and compare

with the Krull-dimension of R. The program we use here is Macaulay 2

Example A.1.1: Let R = k[x3, y3, x2y, xy2]. We compute the depth and

Krull-dimension of R. The algorithm has been properly coded as following,

i1 : k=QQ

o1 = QQ

o1 : Ring


i2 : A=k[t_1..t_4,Degrees=>{3,3,3,3}]

o2 = A

o2 : PolynomialRing

84


i3 : B=k[x,y]

o3 = B

o3 : PolynomialRing

i4 : f=map(B,A,{x^3,y^3,x^2*y,x*y^2})

3 3 2 2

o4 = map(B,A,{x , y , x y, x*y })


i5 : I=ker f

2 2

o5 = ideal (t t - t , t - t t , t t - t t )

2 3 4 3 1 4 1 2 3 4

o5 : Ideal of A

i6 : R=A/I

o6 = R

o6 : QuotientRing

i7 : L=R^1/(module ideal(t_1..t_4))

o7 = cokernel | t_1 t_2 t_3 t_4 |

1

o7 : R-module, quotient of R

i8 : Ext^1(L,R^1)

o8 = 0

o8 : R-module

i9 : Ext^2(L,R^1)

o9 = cokernel {-3} | t_4 0 0 t_3 0 t_2 t_1 0 |

{-3} | 0 t_4 t_3 0 t_2 0 0 t_1 |

2


i10 : dim R

o10 = 2

From the computation, we know that depth(R) = 2 = dim(R). So R is a

Cohen-Macaulay ring. �


Example A.1.2: R = k[x, y, z2, xz, yz] is not a Cohen-Macaulay ring.

i1 : k=QQ

o1 = QQ

o1 : Ring


i2 : A=k[t_1..t_5, Degrees=>{1,1,2,2,2}]

o2 = A

o2 : PolynomialRing

i3 : B=k[x,y,z]

o3 = B

o3 : PolynomialRing

i4 : f=map(B,A,{x,y,z^2,x*z,y*z})

2

o4 = map(B,A,{x, y, z , x*z, y*z})


i5 : I=ker f

2 2 2 2

o5 = ideal (t t - t t , t t - t , t t t - t t , t t - t )

2 4 1 5 2 3 5 1 2 3 4 5 1 3 4

o5 : Ideal of A

i6 : R=A/I

o6 = R

o6 : QuotientRing

i7 : L=R^1/(module ideal(t_1..t_5))

o7 = cokernel | t_1 t_2 t_3 t_4 t_5 |

1


i8 : Ext^1(L,R^1)

o8 = 0

o8 : R-module

i9 : Ext^2(L,R^1)

o9 = cokernel {-1} | t_2 t_1 t_5 t_4 t_3 |

1


i10 : dim R

o10 = 3


The above computation using Macaulay-2 shows that dim(R) = 3, but the

depth= min{i; ExtiR(k,R) 6= 0} = 2, so R is not Cohen-Macaulay. �

A.2. Using the Graded Auslander-Buchsbaum Theorem

This is another easy way to decide if a finitely generated positive graded ring is

Cohen-Macaulay by using a computer. The following statements are well known,

and a reference can be found, for example, in Leslie G. Roberts’ notes [31].

Theorem A.2.1: (Roberts [31], Theorem 13) (graded Auslander-Buchsbaum)

Let R be a finitely generated graded k-algebra in non-negative degrees with R0 =

k a field. Let M be a finitely generated graded R-module with finite projective

dimension, i.e., proj dimRM <∞. Then

proj dimRM + depthRM = depthR

Corollary A.2.2: (Roberts [31], Corollary 14) Let R = k[x1, · · · , xn] and I a

homogeneous ideal in R. Then R/I is Cohen-Macaulay if and only if

proj dimRR/I = n−Krull dimR/I

The above criterion is effective because projective dimension and Krull-dimension

of R/I can be easily found with a computer program.

Example A.2.3: (Example A.1.1) LetR = k[x3, y3, x2y, xy2], A = k[t1, t2, t3, t4].

The computation in the first section shows that R ∼= A/I where I = (t2t3 − t24, t23 −

t1t4, t1t2− t3t4). The projective dimension of R as an A-module is 2 which is shown

by the following computation using Macaulay 2.

i1 : k=QQ

o1 = QQ

o1 : Ring


i2 : A=k[t_1...t_4,Degrees=>{3,3,3,3}]


o2 = A

o2 : PolynomialRing

i3 : B=k[x,y]

o3 = B

o3 : PolynomialRing

i4 : f=map(B,A,{x^3,y^3,x^2*y,x*y^2})

3 3 2 2

o4 = map(B,A,{x , y , x y, x*y })


i5 : I=ker f

2 2

o5 = ideal (t t - t , t - t t , t t - t t )

2 3 4 3 1 4 1 2 3 4

o5 : Ideal of A


o6 = cokernel | t_2t_3-t_4^2 t_3^2-t_1t_4 t_1t_2-t_3t_4 |

1


i7 : res R

1 3 2

o7 = A <-- A <-- A <-- 0

0 1 2 3

o7 : ChainComplex

So R is Cohen-Macaulay by Corollary A.2.2. �

Example A.2.4: (Example A.1.2) Let R = k[x, y, z2, xz, yz], A = k[t1, · · · , t5].We know that R ∼= A/I where I = (t2t4 − t1t5, t

22t3 − t25, t1t2t3 − t4t5, t

21t3 − t24), and

K-dim(R) = 3. By the following computation, we will find that the projective

dimension of R is 3, and since 3 6= 5− 3, R is not a Cohen-Macaulay ring according

to Corollary A.2.2.

i1 : k=QQ

o1 = QQ

o1 : Ring



i2 : A=k[t_1..t_5,Degrees=>{1,1,2,2,2}]

o2 = A

o2 : PolynomialRing

i3 : B=k[x,y,z]

o3 = B

o3 : PolynomialRing

i4 : f=map(B,A,{x,y,z^2,x*z,y*z})

2

o4 = map(B,A,{x, y, z , x*z, y*z})


i5 : I=ker f

2 2 2 2

o5 = ideal (t t - t t , t t - t , t t t - t t , t t - t )

2 4 1 5 2 3 5 1 2 3 4 5 1 3 4

o5 : Ideal of A


o6 = cokernel | t_2t_4-t_1t_5 t_2^2t_3-t_5^2 t_1t_2t_3-t_4t_5

t_1^2t_3-t_4^2 |

1


i7 : res R

1 4 4 1

o7 = A <-- A <-- A <-- A <-- 0

0 1 2 3 4

o7 : ChainComplex

�

A.3. By the Cardinality of the Spanning Monomial Set of a Certain

Semigroup Ring

This algorithm is based on the following theorem in Chapter 3.

Theorem 3.1.12: Let S ⊆ Nn be a simplicial affine semigroup. The set of

spanning vectors contains α1, · · · , αm, and rank(S) = m 6 n. Let the index of the


group G(S) in Zm be h. Then R is Cohen Macaulay if and only if the cardinality

of the spanning monomial set XT is det(B)/h, where B =

α1...αm

. �

Next we will demonstrate how to determine the Cohen-Macaulay property of a

simplicial affine semigroup ring k[S] with S ⊃ G(S)∩◦

C(S) by using the package

TotalProgram under Mathematica environment.

Example A.3.1: Let R = k[x, y, z2, xz, yz].

In[1]:= << "A:\\TotalProgram.m" Load on the package

In[2]:= B={{1, 0, 0}, {0, 1, 0}, {0, 0, 2}} Input spanning vectors

Out[2]= {{1, 0, 0}, {0, 1, 0}, {0, 0, 2}}

In[3]:= A={{1, 0, 1}, {0, 1, 1}} Input

Hilb(S)\{spanning vectors}

Out[3]= {{1, 0, 1}, {0, 1, 1}}

In[4]:= basis[A, B] Calculate spanning

monomial set

Out[4]= {{0, 0, 0}, {0, 1, 1}, {1, 0, 1}}

In[5]:= Length[%4] Find the number of elements

in the spanning monomial set

Out[5]= 3

In[6]:=Det[B] Calculate det(B)

Out[6]= 2

The above computation shows that the number of the elements in the spanning

set is 3, not 1 ∗ 1 ∗ 2 = 2. So by Theorem 3.3.2, R is not Cohen-Macaulay. �

Example A.3.2: Let R = k[x3, y3, xy, x2y, x3y, xy2, xy3].

In[1]:= << "A:\\TotalProgram.m"

In[2]:= B = {{3, 0}, {0, 3}}


Out[2]= {{3, 0}, {0, 3}}

In[3]:= A = {{1, 1}, {2, 1}, {3, 1}, {1, 2}, {1, 3}}

Out[3]= {{1, 1}, {2, 1}, {3, 1}, {1, 2}, {1, 3}}

In[4]:= basis[A, B]

Out[4]= {{0, 0}, {1, 1}, {1, 2}, {1, 3}, {2, 1},

{2, 2}, {2, 3}, {3, 1}, {3, 2}}

In[5]:= Length[%]

Out[5]= 9

In[6]:=Det[B]

Out[6]= 9

We can see that the number of the elements in the spanning set is the same as

det(B), so R is Cohen-Macaulay. �

APPENDIX B

Computer Program - TotalProgram

This program is devoted to supporting the research on the cohomological prop-

erties of semigroup rings in Commutative Algebra, especially focusing on non-

normal semigroup rings. It is written by using Mathematica and works under the

Mathematica environment.

Now this program deals with simplicial affine semigroup rings R = k[S], where

k is a field and S ⊆ Nn is a positive simplicial affine semigroup with S ⊇ C(S)∩Nn.

This program computes,

(1) The interior lattice points of the cone spanned by a simplicial affine semi-

group;

(2) The Hilbert basis of a semigroup;

(3) The spanning monomial set of a semigroup;

(4) Cohen-Macaulay property;

The next four sections will describe how one can compute above properties of

simplicial affine semigroup rings by using the package TotalProgram. In each section

we will provide the mathematical background, algorithm and Mathematica code. We

will also provide some examples to demonstrate a step by step account of how to

use this program.

92


B.1. The Interior Lattice Points of a Cone Spanned by a Simplicial

Semigroup

Let C ⊆ Qn+ be a rational convex cone spanned by m linearly independent

lattice vectors α1, · · · , αm. Let S ⊆ Nn be the affine semigroup generated by all

the relative interior lattice points of the cone C and α1, · · · , αm. Let W denote

the semigroup generated by α1, · · · , αm. Then S = (◦C ∩Nn) ∪W . Clearly S is a

simplicial semigroup and rank(S) = m. We call such a semigroup the semigroup

with spanning vectors α1, · · · , αm. In this section we will put our focus on this type

of semigroups

To understand S, we need to find all the interior lattice points α of C(S) ⊆ Qn

such that α =∑m

i=1 λiαi, λi ∈ Q, λi > 0, for all i.

It is enough to find all the lattice points in the “fundamental region”

F = {α ∈ Nn| α =m∑i=1

λiαi, 0 6 λi < 1, λi ∈ R, for all i}

of the cone C(S) and raise all points of F which are on the (m − 1)-dimensional

faces of C(S) to the “roof” of F except the origin. Here the “roof” of F is defined

as follows,

roof = {α ∈ Nn| α =m∑i=1

λiαi, 0 < λi 6 1, λi ∈ R, and at least one λi is 1}

This set of points is so special throughout the whole program that it is deserved

to be given a name the foundation of the semigroup S, denoted by Γ, i.e.,

Γ = {α ∈ Nn| α =m∑i=1

λiαi, 0 < λi 6 1} ∪ {0}

Here is an example for the case m = n = 2. Let S ⊆ N2 be a simplicial affine

semigroup defined as above. Spanning vectors of the cone C(S) are α1 and α2. Then

the foundation Γ of S corresponds to the large solid dots below.


-

6

b

bbb

bb br rr r r rr r r r r r rr r r r r r rr r r r r r r rr r r r r r rr r r r r r r rr r r r r r rr r r r r rr r r

p pp p pp p pp p pp pp p p p p p p p p pp p p p p p p p pp p p p p p p p pp p p p p p p p p

��

��

(0,0)r

α1

α2

C

Γ

S

B.1.1. Mathematical Background and Algorithm. Take αi as the rows

of the matrix A. A represents a map φ : Nm → Nn. Then for any points α =

(a1, · · · , am) ∈ Nm, φ(α) = αA =∑m

i=1 aiαi ∈ S ⊆ Nn ∩ C(S). So the cokernel of φ

is represented by the set of points in F .

A can be diagonalized, and rank(A) = m since α1, · · · , αm are linearly indepen-

dent. In other words, there are an invertible m×m matrix P and an invertible n×nmatrix Q in Z such that

D = PAQ =

b1 0 · · · 0 0 · · · 00 b2 · · · 0 0 · · · 0

· · · · · ·0 0 · · · bm 0 · · · 0

m×n

is a diagonal matrix, where 0 < b1 6 b2 6 · · · 6 bm, bi ∈ Z, for all i.

Then we have the following commutative diagram,

ZmA

φ // Zn ∩ LQ

��

// coker(φ) // 0

Zm

P

OO

D

ψ // Zn ∩ L′ // coker(ψ)

Q−1

OO

// 0


where L and L′ are linear subspaces of Rn spanned by C(S) and the rows of D

respectively. L′ ∼= Rm. Obviously, the cokernel of ψ contains b1 · · · bm elements

which are represented by all points α = (a1, · · · , am) such that 0 6 ai < bi for all i.

In order to find the cokernel of φ, we simply map coker(ψ) up by Q−1, i.e.,

coker(φ) = coker(ψ) ·Q−1, then translate these representatives of coker(φ) into F to

obtain Γ.

B.1.2. Examples. 1. Let S ⊆ N3 be a simplicial affine semigroup with span-

ning vectors α1 = (3, 0, 0), α2 = (0, 2, 0) defined as before. The cone C(S) ∼= Q2+.

rank(S) = 2. The code for finding the lattice points in the foundation Γ of S is cited

as follows.

In[1]:= << "A:\\TotalProgram.m" Load on the package

In[2]:= A = {{3, 0, 0}, {0, 2, 0}} Input spanning vectors

Out[2]= {{3, 0, 0}, {0, 2, 0}}

In[3]:= integerpoints[A]

Out[3]= {{0, 0, 0}, {3, 1, 0}, {2, 2, 0},

{2, 1, 0}, {1, 2, 0}, {1, 1, 0}

In the above script, the command integerpoints[A] is the command to calcu-

late all interior lattice points in Γ of the semigroup ring spanned by the spanning

vector set A.

2. Let S ⊆ N2 be a simplicial affine semigroup with spanning vectors α1 =

(5, 1), α2 = (2, 4). The points in Γ of S are represented by the solid dots in the

graph on next page.

The lattice points in the foundation of S are obtained easily when we apply the

program TotalProgram.


In[2]:= A = {{5, 1}, {2, 4}}


Out[2]= {{5, 1}, {2, 4}}


Out[3]= {{0, 0}, {3, 4}, {4, 4}, {5, 4}, {6, 4}, {2, 3},

{3, 3}, {4, 3}, {5, 3}, {6, 3}, {2, 2}, {3, 2},

{4, 2}, {5, 2}, {1, 1}, {2, 1}, {3, 1}, {4, 1}}

-

6

bb

bb

r r r rr r r rr r r r r

r r r rp p p p p p p

p p p p p pp p p p p p

p p p pp p p p

p p p

��

(0,0)q (5,1)

(2,4)

C

r

S

B.1.3. Code. Below is the code for the command integerpoints[A] which

is based on the algorithm in this section.

—————————————————————

coker[list_] := Table[i, {j, 1, Length[list]},

{i, 0, list[[j]] - 1}]

cokernel[list_] := Flatten[Outer[List, Sequence @@ coker[list]],

Length[list] - 1]

matrixcokernel[A_] := Module[{m},

m = integerreduce[A];

P = m[[1]];

Q = m[[2]];


D = m[[3]];

cokernel[Tr[Abs[D], List]] . Inverse[Q]]

integerpoints[A_] := Module[{k},

k = matrixcokernel[A] . Inverse[A];

ans1 = (#1 - Floor[#1] & ) /@ k;

Prepend[Rest[ans1] /. {0 -> 1}, ans1[[1]]] . A]

————————————————————–

The command cokernel[list] is to find the cokernel of a map represented

by a diagonal matrix. matrixcokernel[A] is to find the cokernel of the map

φ represented by a matrix A via the cokernel of ψrepresented by a matrix D,

coker(φ) = coker(ψ) · Q−1. integerpoints[A] is to convert the points in the cok-

ernel of the map represented by A into the foundation of S.

Note: integerreduce[A] is a program written by Leslie Roberts which diag-

onalizes the matrix A, i.e., yields invertible matrices P , Q and diagonal matrix D

such that D = PAQ.

Example: A =

(1 2 43 1 2

)We can diagonalize the matrix A by the command integerreduce[A]. The

program used is the following.


In[2]:= A = {{1, 2, 4},{3, 1, 2}}

Out[2]:= {{1, 2, 4},{3, 1, 2}}

In[3]:= integerreduce[A]

Out[3]:={{{1, 0}, {-3, 1}}, {{1, -2, 0}, {0, 1, -2}, {0, 0, 1}},

{{1, 0, 0}, {0, -5, 0}}}


Here P =

(1 0−3 1

), Q =

1 −2 00 1 −20 0 1

, and D =

(1 0 00 −5 0

)such that

D = PAQ.

The following routine gives us an idea of what the results of the commands in

the program look like.


In[2]:= A = {{1, 2, 4}, {3, 1, 2}}

Out[2]= {{1, 2, 4}, {3, 1, 2}}

In[3]:= M = integerreduce[A]

Out[3]= {{{1, 0}, {-3, 1}}, {{1, -2, 0}, {0, 1, -2}, {0, 0, 1}},

{{1, 0, 0}, {0, -5, 0}}}

In[4]:= DD = M[[3]]

Out[4]= {{1, 0, 0}, {0, -5, 0}}

In[5]:= N = Tr[Abs[DD], List]

Out[5]= {1, 5}

In[6]:= coker[N]

Out[6]= {{0}, {0, 1, 2, 3, 4}}

In[7]:= cokernel[N]

Out[7]= {{0, 0}, {0, 1}, {0, 2}, {0, 3}, {0, 4}}

In[8]:= matrixcokernel[A]

Out[8]= {{0, 0, 0}, {0, 1, 2}, {0, 2, 4}, {0, 3, 6}, {0, 4, 8}}


Out[9]= {{0, 0, 0}, {3, 2, 4}, {2, 1, 2}, {2, 2, 4}, {1, 1, 2}}

Remark: The command matrixcokernel gives us representatives of the cokernel

of the map φ. But the points obtained usually are not in Γ. Here the command

“integerpoints” shifts them into the foundation Γ. For example, in the 9th line,


(2, 1, 2) = 15(1, 2, 4) + 3

5(3, 1, 2) ∈ Γ, and (2, 1, 2) = (0, 2, 4) + (3, 1, 2) − (1, 2, 4). So

(2, 1, 2) represents the same element of coker(φ) as (0, 2, 4).

B.2. The Hilbert Basis

B.2.1. Mathematical Background. Let S ⊆ Nn be an affine semigroup.

Recall that the Hilbert basis is the uniquely determined minimal system of irre-

ducible elements of S. If S is positive, then Hilbert basis Hilb(S) is the minimal set

of generators of S.

B.2.2. Algorithm. First of all, we need some more notation.

Let Gens denote the set of non-zero elements of a generating set of an affine

semigroup S ⊆ Nn. Then Hilb(S) ⊆Gens. Let Listq be the set of all sums of

any q elements of Gens. Define Size(α) =∑n

i=1 ai for any α = (a1, · · · , an) ∈ S.

Maxsize(A) and Minsize(A) represent the maximal and minimal size of the elements

in the set A respectively.

Remark:

• 0 /∈ Gens.

• Listq+1 = {α+ β | α ∈ Listq, β ∈ Gens}.• If there is an integer q > 1 such that

Maxsize(Gens) < Minsize(Listq)

then Listj∩ Gens= ∅ for all j > q.

So to find the Hilbert basis of S, it is enough to omit all the points in Listq

(q > 1) from Gens recursively until Maxsize(Gens)< Minsize(Listq).


B.2.3. Code.

mingens[gens_] :=Module[{size, list, mygens, gens},

list = gens; mygens = gens;oursize := Max[(Plus @@ #1 & ) /@ gens]; The Maxsize(Gens)

minsize[list_] := Min[(Plus @@ #1 & ) /@ list]; The Minsize(List).

next[list_] := By remark 2, next[list](xx = Union[Flatten[Table[gens[[i]] + list[[j]], is to find the set of{i, 1, Length[gens]}, {j, 1, Length[list]}], all sums of one element

1] in list and another]; element in Gens.

Select[xx, Plus @@ #1 <= oursize & ] Elements larger than); Maxsize(Gens) cannot be

part of the Hilbertbasis. They are omittedin order to keep thelist from becomingunnecessarily largein the loops.

While[(xxx = minsize[list]) <= oursize, By this recursion, looplist = next[list]; through all list(q),mygens = Complement[mygens, list] q>1, such that]; Minsize(list(q)) is

mygens] less than or equal toMaxsize(Gens). Obtainthe Hilbert basis byomitting any elementin list(q) from Gens.

B.2.4. Examples. 1. Let S be the simplicial semigroup with spanning vectors

α1 = (5, 1) and α2 = (2, 4). A generating set of S, denoted by Gens, can be found

to be the union of two sets of points: the foundation Γ \ {0} of S and the set of

spanning vectors of C. In other words, Gens= {α1, α2} ∪ Γ \ {0}. Then the Hilbert

basis of S contains following lattice points.

Hilb(S) = {(1, 1), (2, 1), (2, 3), (2, 4), (3, 1), (4, 1), (5, 1)}


which are represented by the large solid dots in the following graph.

-

6

b

b

r r r r rp p p prr

p p p p p p pp p p p p pp p p p p p

p p p p p p p pp p p p p p pp p p p p p p

��

(0,0)p (5,1)

(2,4)

C

S

The code of how to use the package TotalProgram to compute Hilbert basis of S is

cited below,

In[1]:= << "A:\\ TotalProgram.m" Load the package

In[2]:= A = {{5, 1}, {2, 4}} Input the spanning

vectors

Out[2]= {{5, 1}, {2, 4}}

In[3]:= Gens = Union[A, Drop[integerpoints[A], 1]] integerpoints[A]

give us elements

in Γ

Out[3]= {{1, 1}, {2, 1}, {2, 2}, {2, 3}, {2, 4}, {3, 1}, {3, 2},

{3, 3}, {3, 4}, {4, 1}, {4, 2}, {4, 3}, {4, 4}, {5, 1},

{5, 2}, {5, 3}, {5, 4}, {6, 3}, {6, 4}}

In[4]:= mingens[Gens] Provide Hilbert

basis

Out[4]= {{1, 1}, {2, 1}, {2, 3}, {2, 4}, {3, 1}, {4, 1}, {5, 1}}


2. Let S be a simplicial semigroup with spanning vectors α1 = (2, 3, 1), α2 =

(0, 3, 2), α3 = (2, 1, 3) The program for finding Hilbert basis of S is used as follows.


In[2]:= A = {{2, 3, 1}, {0, 3, 2}, {2, 1, 3}}

Out[2]= {{2, 3, 1}, {0, 3, 2}, {2, 1, 3}}

In[3]:= Gens = Union[A, Drop[integerpoints[A], 1]]

Out[3]= {{0, 3, 2}, {1, 2, 1}, {1, 2, 2}, {1, 3, 2}, {1, 3, 3},

{1, 4, 3}, {2, 1, 3}, {2, 2, 3}, {2, 3, 1}, {2, 3, 2},

{2, 3, 3}, {2, 3, 4}, {2, 4, 2}, {2, 4, 3}, {2, 4, 4},

{2, 5, 3}, {2, 5, 4}, {3, 4, 3}, {3, 4, 4}, {3, 5, 4},

{3, 5, 5}, {3, 6, 5}}

In[4]:= mingens[Gens]

Out[4]= {{0, 3, 2}, {1, 2, 1}, {1, 2, 2}, {1, 3, 2}, {1, 3, 3},

{1, 4, 3}, {2, 1, 3}, {2, 2, 3}, {2, 3, 1}, {2, 3, 2},

{2, 3, 3}, {2, 3, 4}, {3, 4, 3}}

B.3. The Spanning Set

B.3.1. Mathematical Background. Let S ⊆ Nn be the affine semigroup

with generating set G = {α1, · · · , αm, β1, · · · , βl}, where α1, · · · , αm are spanning

vectors of the cone C(S). Same as before, we write W for the subsemigroup of S gen-

erated by α1, · · · , αm. Recall that the spanning monomial set T of S over W defined

in Chapter 3 is the index set of the k-basis of the quotient ring R/(xα1 , · · · , xαm)R,

or equivalently,

T = {t ∈ S | t− ω /∈ S, for all ω ∈ {α1, · · · , αm}}.

B.3.2. Algorithm. To find the spanning set T of S over W , the algorithm is

quite similar to the algorithm for finding the Hilbert basis which is given in section

2. The only difference is that we need to omit all elements which can be written as

a sum of one element in S and one in W from S.


B.3.3. Code.

minsize[A_] := Min[DeleteCases[Apply[Plus, A, {1}], 0]]

maxsize[A_] := Max[Apply[Plus, A, {1}]]

nextlist[list_, gens_] :=

Union[Flatten[Table[gens[[i]] + list[[j]],

{i, 1, Length[gens]}, {j, 1, Length[list]}], 1]]

basis[A_, B_] :=

Module[{orderA, orderAB, basicorder, a, ab, listofT,

i, j, lastorderAB, dim},

dim = Length[A[[1]]];

orderA = {A};

orderAB =

{Union[Flatten[Table[A[[i]] + B[[j]],

{i, 1, Length[A]},

{j, 1, Length[B]}], 1],

B]

};

lastorderAB = Last[orderAB];

T = {Table[0, {dim}]};

listofT = {T};

x = 0;

While[T != {}, x = x + 1; Print[x];

basicorder =

Flatten[Table[T[[i]] + gens[[j]],

{i, 1, Length[T]},

{j, 1, Length[gens]}],

1];

basicorder = Complement[basicorder, Flatten[orderAB, 1]];

T = basicorder;

While[minsize[lastorderAB] <= maxsize[basicorder],

T = Complement[T, Flatten[orderAB, 1]];


a = Complement[nextlist[Last[orderA], A], Flatten[orderA, 1]];

ab = Union[Flatten[Table[a[[i]] + B[[j]],

{i, 1, Length[a]},

{j, 1, Length[B]}],

1]];

AppendTo[orderA, a];

AppendTo[orderAB, ab];

lastorderAB = ab

];

T = Complement[T, Flatten[orderAB, 1]];

If[T != Last[listofT],

T = Complement[T, Flatten[listofT, 1]];

AppendTo[listofT, T];

]

];

Flatten[listofT, 1]]

B.3.4. Examples. 1. Let S ⊆ N2 be the simplicial affine semigroup with

spanning vectors α1 = (2, 4) and α2 = (5, 1). By the previous calculation in subsec-

tion 2.4, the Hilbert basis is

Hilb(S) = {(1, 1), (2, 1), (2, 3), (2, 4), (3, 1), (4, 1), (5, 1)}

Let W be the semigroup generated by spanning vectors α1 and α2. Then the span-

ning set of S over W is given by the following program.

In[1]:= <<"A:\\TotalProgram.m"

In[2]:= B = {{2, 4}, {5, 1}}

Out[2]= {{2, 4}, {5, 1}}

In[3]:= A = {{1, 1}, {2, 1}, {2, 3}, {2, 4}, {3, 1}, {4, 1}, {5, 1}}

Out[3]= {{1, 1}, {2, 1}, {2, 3}, {2, 4}, {3, 1}, {4, 1}, {5, 1}}

In[4]:= basis[A,B]


Out[4]= {{0, 0}, {1, 1}, {2, 1}, {2, 2}, {2, 3}, {3, 1}, {3, 2},

{3, 3}, {3, 4}, {4, 1}, {4, 2}, {4, 3}, {4, 4}, {5, 2},

{5, 3}, {5, 4}, {6, 3}, {6, 4}}

The command basis[A,B] provides us the spanning set of S over W .

B.4. The Cohen-Macaulay Property

B.4.1. Mathematical Background. Let S ⊆ Nn be the affine semigroup

with generating set G = {α1, · · · , αm, β1, · · · , βl}, where α1, · · · , αm are spanning

vectors of the cone C(S). The Cohen-Macaulayness of such a semigroup ring can

be characterized by the cardinality of its spanning monomial set XT by Theorem

3.1.12.

Theorem 3.1.12: Let S ⊆ Nn be a simplicial affine semigroup. The set of

spanning vectors contains α1, · · · , αm. rank(S) = m 6 n. Let the index of the

group G(S) in Zm be h. Then R is Cohen Macaulay if and only if the cardinality

of the spanning monomial set XT is det(B)/h, where B =

α1...αm

. �

B.4.2. Algorithm. With the techniques above, to find the Cohen-Macaulayness

of a standard simplicial semigroup ring R, we simply compare the cardinality of the

spanning monomial set with the “determinant” of B or D.

B.4.3. Examples. 1. Let S be the simplicial semigroup generated by the

spanning vectors α1 = (5, 1), α2 = (2, 4) and all the interior integer points of the

cone C(S). The index set of the spanning monomial set XT , which is given by the

command basis in section 3, is illustrated by the set of solid dots below.


-

6

bb

bb

r r r rr r r rr r r r rr r r rp p p p p p pp p p p p pp p p p p p

p p p pp p p pp p p

��

(0,0)q (5,1)

(2,4)

C

r

S

Then #XT = 18. det(A)=det

(5 12 4

)= 20 − 2 = 18. So R = k[S] is Cohen-

Macaulay by Theorem 3.1.12.

2. Let S be the simplicial semigroup with spanning vectors α1 = (3, 0, 0), α2 =

(0, 2, 0). The cone C(S) ∼= N2 is the xy coordinate plane. S contains all the relative

interior integer points of C(S). The following code is cited for finding the Cohen-

Macaulayness of R = k[S].


In[2]:= B = {{3, 0, 0}, {0, 2, 0}}

Out[2]= {{3, 0, 0}, {0, 2, 0}}

In[3]:= R = Union[B, Drop[integerpoints[B], 1]]

Out[3]= {{0, 2, 0}, {1, 1, 0}, {1, 2, 0}, {2, 1, 0}, {2, 2, 0},

{3, 0, 0}, {3, 1, 0}}

In[4]:= T = basis[R, B]

Out[4]= {{0, 0, 0}, {1, 1, 0}, {1, 2, 0}, {2, 1, 0}, {2, 2, 0},

{3, 1, 0}}

In[5]:= Length[T]

Out[5]= 6

In[6]:= DD = integerreduce[B][[3]]

Out[6]= {{3, 0, 0}, {0, 2, 0}}

Appendix 2. Computer Program - TotalProgram 107

In[7]:= det(DD)= Tr[DD, Times]

Out[18]= 6

-- Now the length of T is equal to the determinant of DD,

so R is Cohen-Macaulay.

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Curriculum Vitae

Name

Ping Li

Education

Ph.D. Mathematics and Statistics DepartmentQueen’s University, September 2000 - Expected,Research Area: Commutative Algebra and AlgebraicGeometrySupervisor: Leslie G. Roberts

M.Sc. Mathematics Department, Hebei Teacher’s UniversityHebei, China. Sep. 1997 - July 2000Research Area: Differential and Algebraic TopologySupervisor: Zongze Liu

A Diploma in Mathematics Department, Hebei Teacher’s UniversityComputer Science Hebei, China, September 1993 - July 1996

B.Sc. Mathematics DepartmentHebei Teacher’s University, September 1985 - July 1989

Work Experience

9/2000 - present Research AssistantMathematics and Statistics DepartmentQueen’s University

9/2002 - present Teaching Fellow (Instructor for Math 126)Mathematics and Statistics DepartmentQueen’s University

9/2000 - 8/2002 Teaching Assistant (Marker for Math 006, 218/318,tutor for Math 126)Mathematics and Statistics DepartmentQueen’s University

9/1989 - 8/2000 LecturerFaculty of ScienceHebei Science and Technology University, Hebei, China

112

Courses Taught:

Math 126 (Calculus)

Fall and Winter term 2004-2005; Fall and Winter term 2003-2004,

Fall term, 2002

Mathematics and Statistics Department, Queen’s University

Linear Algebra, Computer Language, Calculus

9/1989 – 7/2000

Faculty of Science, Hebei Science and Technology University, Hebei,

China

Awards

Queen’s Graduate Awards Queen’s University, 2000–2004

113

seminormality and the cohen-macaulay propertypingli/thesis.pdf · in the early 1970’s , hochster...

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