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1969 Generators and Relations of Abelian Semigroups and Semigroup Rings. Juergen Reinhard gerhard Herzog Louisiana State University and Agricultural & Mechanical College
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HERZOG, Juergen Reinhard Gerhard, 1941- GENERATORS AND RELATIONS OF ABELIAN SEMIGROUPS AND SEMIGROUP RINGS.
The Louisiana State University and Agricultural and Mechanical College, Ph.D., 1969 Mathematics University Microfilms, Inc., Ann Arbor, Michigan GENERATORS AND RELATIONS
OF ABELIAN SEMIGROUPS AND SEMIGROUP RINGS
A Dissertation
Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Ph i1osophy
in
The Department of Mathematics
by Juergen Reinhard Gerhard Herzog Vordiplom, Universitaet Heidelberg, April, 1966 M.S., Purdue University, June, 1968 August, 1969 ACKNOWLEDGMENTS
The author wishes to express his gratitude to Professor Ernst
Kunz for his guidance and assistance in the preparation of this dissertation, and especially for suggesting the questions studied here. Thanks are also due to Professor Otto F. G. Schilling, who was responsible for financial support for the author in the
summer of I968 at Purdue University where part of this dissertation was written. Last not least, the author expresses his appreciation
to Mr. Klaus Nagel, whose knowledge in programming enabled the
checking with a computer of some conjectures on numerical semigroups.
i i TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ''
ABSTRACT iv
INTRODUCTION vi
CHAPTER I. Monomial Varieties 1
1.1. Basic Definitions and some Geometric Properties 1
of Monomial Varieties.
1.2. An Isomorphism Theorem 5
CHAPTER II. Connections between Semigroups and their Associated
Semigroup Rings. 10
2.1. The Defining Relations of a Semigroup and its
Semigroup Ring. 10
2.2. Complete Intersections. 15
CHAPTER III. Sylvester-Semigroups and their Semigroup Rings. 20
3.1. Properties of Sylvester-Semigroups. 20
3.2. The Semigroup Ring of a Sylvester-Semigroup. 2k
CHAPTER IV. Numerical Semigroups of Rank 3. 30
k.\. The Minimal Relations of Numerical Semigroups of
Rank 3. 30
k.2 Equivalent Conditions for complete intersections. 41
i i i ABSTRACT
The object of this paper is to find necessary and sufficient conditions for an affine space-curve C with parametric equations ni n0 n, x = t,y= tz, z a t- 3 (n1 ,n2,nj positive integers with g.c.d.(nj ^ 2 ^ ^ ) = 1) to be an ideal theoretic complete intersection.
In Chapter I we study algebraic varieties, whose coordinate ring is the semigroup ring k[S], where k is an algebraically closed field , and S designates a geometric semigroup. A semigroup S is said to be a geometric semigroup if S is a finitely generated subsemigroup with 0 element of a finitely generated free abelian nl n9 group. The curve C = {(t , t , t )/ t€k} is a special case of these varieties. We prove the theorem that if R is an integral domain, S a semigroup of integers, and S' any finitely generated semigroup, then R[S] =* R[S'] implies S = S '.
In Chapter II we give a new proof of the fact that each finitely generated abelian semigroup is finitely presented.
Moreover,«we show that if S is a geometric semigroup with no
invertible elements, then the number of relations defining S is greater than or equal to the least number of generators of Sminus the rank of the associated group of S. If equality holds, we say that S is a complete intersection. We close this chapter by proving that S is a complete intersection if and only if the variety belonging to k[S] is an ideal theoretic complete intersection.
Chapter III is devoted to the study of Sylvester-semigroups and their semigroup rings. We call a subsemigroup of the natural numbers a Sylvester-semigroup if there exists an integer m such that zcS if and only m-ztfS, for all integers z. We show that a semigroup S is a Sylvester-semigroup if and only if the localization
0Q of the semigroup ring k[S] at the origin is a Gorenstein ring.
It turns out that the ideal theory of Sylvester-semigroups is similar to the ideal theory of Gorenstein rings. Using a result of Serre we conclude that a semigroup of natural numbers generated by 3 elements is a complete intersection if and only if S is a
Sylvester-semigroup.
In Chapter IV a direct proof of this theorem is given. We also give two other equivalent conditions for a semigroup of natural numbers generated by 3 elements to be a complete intersection.
v INTRODUCTION
This study will present necessary and sufficient conditions for a certain class of rational varieties to be ideal-theoretic complete intersections. The beginning of the investigation will be a classical example of a curve in the affine 3-space, which is
not an ideal-theoretic complete intersection: this curve is given by the parametric representation
C = {(t3,t2*,t5) c k3/ tek} where k is an algebraically closed field. The coordinate ring of
C is k[T3 ,T^,T3] where T is an indeterminate. The ring k[T3,T^,T^] may be considered as a semigroup ring k[S] where S is the semi
group generated by {3,^,5}. This point of view suggests the following generalization: Let S be a finitely generated subsemi
group (with 0 element) of a finitely generated free abelian group. Such a semigroup will be called a geometric semigroup. Let lg be the kernel of the canonical epimorphism
k[Xp...,X ] -* k[Sj, the X.'s being mapped onto the generators of S. Let VQ be the affine variety in the affine n-space A (k) defined by I I n other words:
V$ = {*eAn(k) / F(x) = 0 for all Felg]
We call Vg the monomial variety defined by S. These are actually the varieties we are interested in; the curve C is a special case of them.
vl In the first section of Chapter I we summarize some geometric properties of monomial varieties and turn, in the following section, to the question: If S and S' are geometric semigroups and Vg, Vg, their affine varieties over krespectively, can Vg and Vg, be bi- regularly equivalent with S and S' non-isomorphic?
Concerning this question, we prove that if R is an integral domain, S a subsemigroup of the natural numbers, and S' any finitely generated semigroup, then
R[S] sr R[S'] implies that S » S'
The proof of this theorem was indicated to the author by E. D. Davis.
In Chapter II we describe the ideal |g in detail. It turns out that the generators of lg are essentially given by the generators of the semigroup S. This fact will enable us to give a simple proof of the theorem that each fin itely generated commutative semi group is finitely presented, (cf L. Redei, [4]). In fact, by our method, this is a special case of Hilbert's basis theorem.
At this point, the question of how many relations are necessary to define a given semigroup arises. We prove that if S is a geometric semigroup, then the least number of elements of a set of defining relations for S is greater or equal to the least number of generators of S minus the rank of the associated group § of S. If equality holds, we say that S is a complete intersection. We conclude this section by proving that if S is a geometric semigroup with no invertible elements, then Vg Is an Ideal-theoretic complete intersection if and only if S is a complete intersection. This
v i i theorem reduces the question of complete intersections for monomial varieties to the similar question for the defining semigroups. More over, we obtain as a corollary that Vg is an ideal-theoretic complete intersection if and only if Vg is locally an ideal-theoretic complete intersection.
These results are still somewhat unsatisfactory, becquse, in general, it is a hard problem to decide whether a given geometric semigroup is a complete intersection or not. Even if we are dealing only with subsemigroups of the natural numbers (i.e. numerical semigroups), this problem is so far unsolved. However, it is easier to determine those numerical semigroups whose semigroup ring k[S] is contained in a larger class of rings which are locally
Gorensteinrings.
In Chapter III we investigate the special case where S is a numerical semigroup. In the firs t section we compile some known facts about numerical semigroups and introduce the notion of
Sylvester-semigroups. We say that a cancellative semigroup S is
A .A a Sylvester-semigroup if there exists an element x e S (S is the associated group of S) such that for all elements s e § the following hoids:
(*) s e S if and only if x-s £ S
(S is understood to be embedded in S*)
This notion was introduced by R. Apery in his paper [1 ].
He calls a semigroup with the above peoperty "symmetric". According
v iii to L. R^dei, Sylvester proved that each numerical semigroup which is
generated by two relatively prime elements has the property (*).
Since the term "symmetric" is already used in semigroup theory in
other contexts, we prefer to call such a semigroup a Sylvester-
semigroup. In the following section of Chapter III, we show that the localization at the origin of the semigroup ring k[S], where k
is a field and S a numerical semigroup, is a Gorensteinrihg if
and only if S is a Sylvester-semigroup. This theorem suggests
that the ideal theory of a Sylvester-semigroup should be similar
to the ideal theory of a Gorensteinring. We show that this is
the case. With the aid of this theorem and a result of Serre [5],
we conclude: A numerical semigroup of rank 3 is a complete inter
section if and only if it is a Sylvester-semigroup.
In Chapter IV we prove this theorem directly and give some
equivalent conditions for numerical semigroups of rank 3 to be
complete intersections. The main tool of the proof will be the
notion of a minimal relation. Finally, we show by examples that
many of our results do not hold for semigroups generated bymore than
3. elements.
ix NOTATIONAL CONVENTIONS
The positive integers will be denoted by N; the positive integers together with 0 by N^; and the integers by Z. If E n is a set, En will always denote ff Ej where Ej = E for i=l i = l , . . . , n . If R[S] is a semigroup ring then the elements of the natural basis of R[S] over R will be denoted by xg, scS.
x CHAPTER I
Monomial Varieties
1.1. Basic definitions and some geometric properties of monomial
variet ies.
As indicated in the introduction we will study a certain class of
rational varieties. To describe them we need some definitions.
Definition 1. i . 1.: S is a geometric semigroup if S is a finitely
generated subsemigroup with 0 element of a fin itely generated free
abei ian group.
The reason why we call such a semigroup a geometric semigroup,
is that the reader may visualize it as a subsemigroup of the groups of lattice points of Rn, n suitably chosen.
Let k be an algebraically closed field.
Definition 1.1.2.: An algebraic variety VS Am(k) is called a monomial variety, if its coordinate ring k[V] = k[Xj, *
I(V) = [f € k[X1...... Xm]/f(x) = 0 forall x e V} is isomorphic to
the semigroup'ring k[S] over k of a geometric semigroup S.
For a geometric semigroup S there is a canonical construction of a
monomial variety VQ : The associated group 3 of S is a finitely
generated, free abelian group and, since S is cancel 1 ative, we have
an embedding S-*§. We will identify S with its image in S. Also,
since*? is a finitely generated free abelian group, we may identify
S with Zn for some positive integer n. Thus we may assume that
1 S £LZn and 3 = Zn. Let {vj»-**»vm3 s z° be a minimal set of generators of S. Say v. = U j j ) , ' = 1. • • • »m.J ■ 1 n» Let k be an algebraically closed field. Let Tj ,.. fT be indeterminates and v. n i.. T ' = ft T , i = l,...,m . It is clear that the semigroup ring j=l n V1 Vm krs] is isomorphic to k[T , . . ,T ] . We have an epimorphism £ from the polynomial ring k[Xj,...,X ] in m-varlables onto the ring . v, v v. k[T .. .T m] defined by : £ (X.) * T , i= l m . Let lg be the kernel of £, . We define:
Vs - V 0S)
(V(ls) = {x c Am(k)/f(x) ■ 0 for all f e lg})
We should keep in mind that the definition of Vg depends on the identifications we just made.
From the definition of Vg follows: a) Vs S A m(k) b) The coordinate ring of Vg is k[T \ ...,T m] c) The function field of Vg is k(Tj,.. . »T*n) d) Vc is an irreducible rational variety e) dim Vg = n = rank^ S
Let H., i = l,...,m be the hyperplanes in Am(k) defined by the equations" {X. = 0}._, i i** I y • ^ni
The following theorem summarizes some properties of monomial varieties: Theorem 1.1.3.; With the above assumptions we have: v, v 1) If x ( VgH C(H|U...UHm) then x Is of the form x » (t t )
v. n 2|J where t * tr t , J for I * 1 ,... ,m and with t ,ck for j ■ 1,.... ,n. j-l J J 2) The singular locus of Vg is contained in
V$r\ (H^u
3) The origin Is a point of Vg if and only if S has no Invertible elements.
4) I f m > n and If the origin Is a poInt of Vg, then 11 is a singular point.
Proof: 1.) Let x € VgO C(H|u . To x corresponds a k-algebra homomorphism
vl v Let R - k[T T m]rTVi be the quotient ring of 11 V€S v. v k[T ,...,T m] with respect to the mul t ipl icat i vely closed set fr’lvrt then R " kl-Tl Tn^tTV5vfzn This Is a special case of a general fact: Lemma 1.1 .k.i If R Is an unitary ring, S a commutative semigroup, If : S-& the natural morphism of S into its associated group § and R[ff] R[S] -* r[§] the ring homomorphism induced by ff, then R[§] is isomorphic to R[S]£Sj , the quotient ring of R[S] with respectto {S} and R[ir] is the natural mapping of R[S] into its quotientring Since x | H, U...U H we can 1 ift <0 to * I m x *x V1 1 k[T m] k is commutative. Say ? x(Tj) = t . , t. € k then 2.) Let the kernel of 3.) If S has invertible elements, then there exists an equation m v* V' iff (t 1) 1 = 1, V. ^ 0, V. integers. So the origin cannot be a i=l 1 1 point of Vc. Conversely, if S has no invertible elements, then y V. V V. V (T , . . . T m) is a maximal ideal in k[R ,...T ] corresponding to the origin. k.) If m> n then lQ is generated by polynomials of the form t m m ^ j ^ m IT X. - it X. with L v. > 1 and X fi. > 1. (We will prove i=l 1 1=1 1 i=l ' i=l 1 this in the,next chapter). If we take this for granted, then it is clear that the Jacobian of Vg vanishes at the origin, so that in fact the origin is a singular point. With the same assumptions we get: Corollary 1.1.5.: V„ has outside of H U...U H_ fhe parametric d j . . ».ni representat ion C(t 1,...,t m)c Am(k)/t. € k, t. f 0, i = 1...... n}. Let S be a numerical semigroup, (i.e. S is a subsemigroup of Nq). Say S is generated by n j,f..,n^ € N and {n^ ...... n^] is a minimal system of generators of S. Then we obtain as a special case: Corollary 1.1.6.: Vg is a curve in A^(k) passing through the origin and having there its only singular point. Moreover, we have for VQ the parametric representation w Vg = C(t 1,... ,t A^(k) / t € k} Remark: The parametric representation doesn't hold in general gl obal1y. Example: Let S be generated by Vj = 0 , 0 , V2 = (1,2), v^ = (1,3) then Vg = C(x1,x2,x3) c A3(k) / x]x3-x2 = 0}. Therefore (l,0,0)e Vg, but (1,0,0) C(tr t2, y t* , yt;*) / t,,t2 C k} 1.2. An isomorphism theorem Before we turn to the intersection problem we are going to prove a theorem, which is of a certain interest in this context. Let S, S' be geometric semigroups and Vg, Vg' their affine varieties over k. Can Vg and Vg, be biregularl-y equivalent for non-isomorphic S, S'? More generally the question 6 can be stated: If R Is an arbitrary commutative, unitary ring, and S,S‘ arbitrary commutative semigroups with 0 element, does an R-isomorphism of the semigroup rings R[S], R[S'] imply that S and S' are isomorphic? Before proving the main theorem, we show: Proposition 1.2.1.: For an abelian semigroup S the following conditions are equivalent: a.) S is a geometric semigroup b.) 1.) S is finitely generated and has a 0 element 2.) Sis cancellative 3.) Whenever n s^ = n for n c N and S j,s2 e S then Sj = s^ c.) 1.) S is finitely generated and has a 0 element 2.) If R is an integral domain then R[S] is an integral domain. Proof: a.) implies b.) is triv ial and a.) implies c.) is clear, since R[S] can be considered as subring of a polynomial ring R[X, X ]_, local ized with respect to T = {XV]! (Lemma 1.1.4.). 1 n T veN" b.) implies a.): Denote the associated group of S by S. There is a canonical homomorphism ff : S -* 3. By assumption S is cancel 1 ative, therefore ff is an embedding. Since S is finitely generated, ? is also finitely generated. It remains to show that § is free or equivalently; 3 is torsion free. Consider the equation ri! = 0 with neN and *§6^. We have to show that ^ ■ 0. ^ can be written: “s = ff(sj) - ff(s^) where Sj , s2cS. Therefore 0 = ff(nsj) - ff(ns2) , thus ir(nsl) = ff(ns2) . Since ir is injective this implies that nSj = ns^ and by condition 3.) Sj = s2, therefore £ «= 0. c.) implies b.) : We only have to check 2.) and 3.). 2.) Assume S is not cancellative, then there exist s,Sj,s2eS s, ^ 52 such that s, + s = s2+s. Then ( x ^ - x ^ ) x s - but x - x j4 0 and x ^ 0, a contradiction., Sj Sg s I 3.) Assume there exist Sj .s^eS such that Sj?4 s2 and n-l ns, - ns2 for some ncN. Then (*S]- x(„-v)s,+vs2>= xns, n*l but x ■ x J O and S x,„ * j4 0, a contradiction. sl 2 v=l (n-v)s,+vs2 Theorem 1.2.2.; Let R be an integral domain, S a numerical semigroup, and S' any finitely generated semigroup with 0 element Then R[S] = R[S'] implies that S “’S'. R Proof: Let k be the quotient field of R then also k[S] ~ k[S'] . Since k[S] is an integral domain, so is k[S‘] and by proposition 1.2,1. S' must be a geometric semigroup. Since dim k[S] = I also dim k[S'] =1. In section 1.1., we showed that dim k[S']= . hence, S' = Z and therefore S' & Z. Assume S' has invertible elements, then S' ~ Z and the elements xgek[S[], seS', are units in k[S']; whereas the units of the ring k[S] are just the elements of k, different from zero, a contradiction. Thus S' is isomorphic to some numerical semigroup. Theorem 1.2.2 is therefore reduced to the special case (E. D. Davis): If k is a field and S, S' are numerical semigroups such that k[S] =* k[S'], then S ~ S ‘. Proof: i.) I f S = N then k[S] is isomorphic to the polynomial ring in one variable k[T] and therefore k[S] = k[T], Assume S' is a proper subsemigroup of N. Say S' is generated nl I by Cni> n 2 ’ ,nP » n'| e N* Then k[S'] = k[T 4] and we know that k[T ni T ni ] m ntf is a non-regular local ring. (T T Z) However all localizations of the polynomial ring with respect to prime ideals are regular local rings, therefore k[S] cannot be isomorphic to S ', unless S' is also jsomorphic to N. 2.) We may suppose that both S and S' are proper subsemigroups of Nq. Say S is generated by {rip.. ..n^}, n.cN. We may assume that the greatest common divisor of npn2,...,n^ is 1, because ~ nl n2 nA if v = gcd(n, ,n2,;..,n^) then?*, generated by , is isomorphic to S. Let S' be generated by {nj nj^}, n.^N . We also assume that gcd(nj,... ,nj^) = 1. We get n. n„ n! n' k[T ' T *] ~ k[S] = kCS'3 - k||T , . . . ,T k]. n, n. nj nk Let points of the corresponding varieties. The integral closure of n, n. ni . n' k[T X] as well as of k[T *"} is k[T]. Therefore k[T] 2 k[T] n . nj J n { n l k[T T ] - k[T ,...,T ■ ] By Luroth's theorem we get p(T) = a + £T with a, j8ck and P £ 0. Since = ^ ; we obtain our case « must be zero. Thus $>(T) = /f*T with £ck, P £ 0. Since ~ n? is the restriction of p to k[T T ], we have all seS, thereby proving that S = S'. Corollary 1.2.3.; If k is a field andS a numerical semigrpup then Aut^(k[S], k[S]) =* k*. Remarks: The proof of Theorem 1.2.2. cannot be generalized to arbitrary geometric semigroups, because the integral closure of k[S] need not to be a polynomial ring. Even if this is so, things are much more complicated, since the automorphism group is in general not as trivial as in the case of corollary 1.2.3. Example: We define an automorphism «P Connections between semigroups and their associated semigroup rings 2.1. The defining relations of a semigroup and its semigroup ring. First of all we introduce some notations and definitions and recall some well-known lemmas. Definition 2.1.1.: S is a free semigroup of rank n if there exist n n s. ,... ,s eS such that any equation £ v.s. = £ v.'s. with v.,' v.eN, 1 n j=l i=l i = l,...,n yields v. = v.' for all i= =» l,...,n. Free semigroups have the following universal mapping property: Let F be a free semigroup of rank n and S a commutative semigroup. Assume F is generated by {ej,...,e }.' Pick any elements S j,...,sneS Then there exists exactly one morphism As corollaries we get: 1.) Any two free semigroups of the same rank are isomorphic. 2.) Any fin itely generated commutative semigroup is an epimorphic image of a free semigroup. Now let S be a finitely generated commutative semigroup with 0 element. Then there exists an epimorphism p : F -* S, where F is a free commutative semigroup. The morphism p defines a (binary) relation on F: p = C(v,w) eFxF / p*(v) = p*(w)) 10 Obviously p is an equivalence relation and moreover has the following compatibility property C : If (v,v') cp and weF then (v+w,v'+w) ep. According to Redei we define: Definition 2.1.2.: A subset p £ FxF is called a congruence on F if l.).p is an equivalence relation 2.) p has the property C. If,p is a congruence on F, then the set of equivalence classes F/p can be given a semigroup structure in an obvious way and there is a natural epimorphism p : F -* F/p. Lemma: If F and S are finitely generated commutative semigroups, F is fre e ,,p : F -* S an epimorphism and p £ FxF the congruence p = {(v,w) jfFxF / p (v) = p (w)} then F/p is isomorphic to S. Hence any finitely generated commutative semigroup can be written as a free semigroup modulo a congruence. The trivial congruence on F is i = £(v,v) eFxF/vcF) and clearly F/i = F. For any relatioh p on F we put p {(v,w)cFxF/ (w,v)cp}. It is rather obvious that for an arbitrary relation ,p on F there is a smallest congruence p containing p, namely the intersection of.a l1 congruences containing p. There is an explicit construction of p : 1. Step: Put po = p u p ~ \ i, then pQ is reflexive and symmetric and p c. pQ . 2. Step: Put P| = {v+w, v'+w)€FxF/(v,v')epo wcF], then p, is reflexive, symmetric and satisfies condition C and p £ pQ £ Pj. 3. Step: (Transitive closure) Define (v,w)cFxF to be an element of p if there exist vq ,Vj, . . . .v^eF, vq = v v^ = w with (Vj, v.+j)cp, for al1 i = 0,1,... , j& - 1 . A simple verification shows that p is in fact the desired congruence. Definition 2 1.3.: A congruence p on F is said to be finitely generated if there exists a finite subset cr s P such that a = p. For a fin itely generated commutative semigroup S we define: Definition 2.1.A.: S is finitely presented if S F/p, where F is a free semigroup and p a finitely generated congruence. We now prove a representation theorem for semigroup rings and introduce for this purpose some notations: Any free semigroup F of rank n is isomorphic to NQn = {(v1 vn) / v.cNq, i=l n) with addition componentwise, and any commutative semigroup S of rank n is an epimorphic image.p'? : Nq0 -• S and S HQn/p, where ☆ p is the congruence determined by p Let R be a commutative, unitary ring and R[Xj Xn] be the polynomial ring in n variables over R. If vcNq0 , v = (V|,...,vn), n v. we define XV = n X. 1 . R[X, X ] is isomorphic to i=l i f n R[N0n] by XV The epimorphism p*: NQn -* S induces an We may consider R[Xj,...Xn] as an S-graded ring in the following sense: A polynomial FeR[Xj,. . . ,X ] is said to be-homogeneous of _ \/ Vc degree s, seS if F = 35 r X with p (v) = s for all s such that y€N0nV rv £ 0. Also R[S] is in a triv ial way an S-graded ring. From the definition of R[p ] follows, that R[p ] is a homogeneous homo- morphism of degree 0. We denote the kernel of R[p ] by lg and for Aep, A = (v,w), v,weNQn we put f a = XV - x w Theorem 2,1.5: lg = ( l ^ ^ p ) . * In other words: R[S] = R[N P/p] ~ R^N0 ^//cc •> v 0 ' 1 AJAcp; Proof: Let J = (CFA3Aep) . 1.) J £ lg, clearly by the definition of R[p ] 2.) Observe that lg is a homogeneous ideal in the S-graded ring R[X|,. . ,X ] . Therefore it suffices to prove: If Fel«. and F is homogeneous, then Fc'J. Choose Fclg, F homogeneous of degree s. m-1 v. v m-1 F = L r (X -X m) = S r FA , where A. = ( v .,v j for i=l vi i=l v i Mi i i m ft ft i=l , . . . ,m-l. Since p (v.) = p (vm) for al 1 i=l m-1, we get that A.cp for i= l ...... m-1, thus Fe'J . Remark: This theorem can easily be generalized to commutative semlgroupsS, which are not necessarily finitely generated. With the same assumptions as for Theorem 2.1.5 wehave the following: i Proposition 2.1.6.: Let a = £A}...... Am}, where A.ep for i=1 ...... m. Then the following conditions are equivalent: 1.) Cr => p 2.) I g = (Fa »• • • *Fa ^ 1 m Proof: 1.) implies 2.): a.) Let p0 = a KJ a -1u i, Jrhen ( tFA^Ac<0^) s ^FA1,,,,,F A ThiS ls triv ia l, because if Acer"1, then A => A. 1 for some !=!,.►. ,m and FA = F ,= - F ., and if Aci then F. = 0. A7 Ai A i b.) Let p, = {(v+w, v'+w)/(v,v')ep0 , wcNg"} then (£FA3Ae^ ) = ^ FA^Aep because then there exists wcNq11 and A'cPq such that F. = XWF , . A A1 c.) Let Acp, A = (v,w). By assumption a = p, and we know that O' is the transitive closure of Pj. Therefore there exist BQ,... ,B( B. = (v.,w.) such that v = vn , w = w. and v .fl = w. for i =» 0 k-1 . i ' i * i' 0 k i+l i k Hence F..= 2 F . a.), b.) and c.) together establish our assertion. A i=0 Bi 2.) implies 1.): Nn We know already that (F^ ,..,,F A ) = (£FA3A ")• Let s ‘ = and it 1 m cr : Nq -* S’ be the canonical epimorphism then, since lgt = (£FA}Af“) 0 -» (Fa , . . . ,Fa ) -• R[X1,... ,Xn] ^ R[S] -* 0 is exact. By 1 m assumption lg = (Fa ,...,Fa ) = lg, . Therefore, if Acp, A = (v,w), , 1 m then R[ Proof: An easy consequence of 2.1.5., 2.1.6. and H ilbert's Basis Theorem. Remarks: R^dei proved this fact in his book [4] and the proof also appears in [3]. 2.2. Complete intersections. In this section we prove a theorem on the number o f relations for geometric semigroups and introduce the notion of "complete .intersections" for semigroups. Let S be a geometric semigroup. A Definition 2.2.1.: dim S = : rank., S. The following fact gives a justification of this definition: If k is a field then (Krull) dim k[S] =» dim S (See Chapter I). The next two definitions refer to arbitrary finitely generated semigroups S, Definition 2.2.2.: The rank of S is the number of elements of a minimal system of generators of S. If p S F is a congruence we define: Definition 2.2.3.: The rank of p is the number of elements of a minimal system of generators of p. Theorem 2.2.^.: Let S be a geometric semigroup and represent S as S » F/p, where F is free and p is a congruence. Then 16 rank p S: rank S - dim S. Proof: We make use of the following theorem: If k is a field, k[X ,....,X ] the polynomial ring in n-variables over k, I n ^ a ({Fj,...,F^}) i deal in ktXj,...»X^] and A a kCX j , •. • (X^] y ^ then r ^ n - dim A. In our case we have: dim S ». dim k[S] = dim (k^F^ //r e \ x) . If rank p = r then there K A^Aep exist A p ...,A rep such that (£FA}Ae/Q) = ({Fa F )), see 2.1.6. 1 Ar We get therefore: rank p = r = dim k[F] - dim k[S] = dim F - dim S 2: rank S - dim S, since rank S s dim F; This proves our theorem. Let S be a geometric semigroup. Definition 2.2.5.: S is a complete intersection if S can be represented as S a F/p, where F is free and p is a congruence, such that rank.p = rank S - dim S. As a straightforward application of the next theorem we will find that a geometric semigroup with no invertible elements is a complete intersection if and only if the monomial variety Vg defined by S is an ideal theoretic complete intersection. Let R o 0 R be an S-graded ring, where S is a geometric semigroup. sjS s From now on we will assume that S has no invertible elements then p = Gt R becomes an ideal in R. If/0V£R is an arbitrary seS s ss^O finitely generated ideal in R, we define the rank o f^ to be the least number of generators of 'tfc . Theorem 2.2.6.: Assume RQ is an integral domain. Let/fZ. be a fin itely generated homogeneous ideal in R and assume that 4)1/is generated by the homogeneous elements Cap.**»an3 and rank HI = m. Then there exist a. , ...,aj c{a| ...... a^} such that I m = (a j a * ) . 1 m In other words: We can find -a' minimal system of generators of ^ in a set of homogeneous generators of M> . Proof: The ideal P = ® Rg is a prime ideal in R, since RQ i s an s^O integral domain. By assumption rank'Ol- = m,' therefore we can choose a a. } such that til, R = (a a ), where the *1 'm 1 n K 'l 'm a. 's are the images of the a. 's in R . This can be done, since .j 'j in a local ring one can select a minimal system of generators of an ideal from any system of generators. We show now that {a. ,...,a. ) * * y1 I m is a global system of generators of 4)V . Since is homogeneneous weuonly have to.prove: . If a e ^ a lid a is homogeneous, then a;e(aj.,.... ,a. ).. We can certainly find r. eR, 1 n • 1 •' i =...0, 1 *.■. 1%. i P such that r a = S r. a. . We may assume U • * I I * J=l J J . >n that rA = 1 -r_', r ' e P. Therefore, a => r 'a + E r. a. . 0 0 ’ 0 0.,ji. 18 For an element reR we denote the homogeneous component of degree s of r by h (r). The element a is assumed to be homogeneous. Say, s m deg a = s. Then a ■ h_(r'a) + h ( E r.a. ), but h (r'a) => 0, 5 U S ( « J I i S U m since r 'eP. Therefore a = h ( E r. a. ) . 0 S j=l J 'j However, the ideal (a. , .. ., a . ) is homogeneous so that the last 'l 'm equation yields ae(a. ,. .,a. ) . 'l ’ 'm Let S be a geometric semigroup with no invertible elements. Represent S as F/p and let lg be the kernel of R[p ]:R[Xj Xn] -* R[S]. As a corollary of 2.2.6 we get: Corollary 2.2.7.: rank p = rank lg . Proof: Obviously rank p ^ rank lg, (see 2.1.6). For the converse remember that R[Xj Xn] is an S-graded ring in the sense of section 2.1. and l=({FA3A ) is a homogeneous ideal in 5 A Acp R[X j,...,X n] generated by the homogeneous elements CF^}A€p The inequality rank p £ rank lg follows now from 2.2.6 and 2.1.6. Corollary 2.2.8.: Let S be a geometric semigroup with no invertible elements. The following conditions are equivalent: a.) S is a complete intersection. b.) V.g is an ideal theoretic complete intersection. c.) Vg is locally an ideal theoretic complete intersection. d.) Vg is locally at the origin an ideal theoretic complete Intersection. Proof; It Is trivial that a.) implies b.), b.) implies c.) and c.) implies d.). We only have to show that d.) implies a.),- but this is a simple consequence of 2.2.7 and 2.1.6. Let lgW =* {deg /- Acp}. Obviously lg* is an ideal in S. If S is a geometric semigroup with no invertible elements, whose rank is m and dimension is n, then we get the following necessary condition for Vg to be an ideal theoretic complete intersection. Corollary 2.2.9.: If Vg is an ideal theoretic complete intersection, * then rank lg £ m - n. Proof: If V„ is an ideal theoretic complete intersection, then rank p M-!m-n for a suitable representation F/p of S. Obviously rank l$ s: rank l$ . But rank lg p rank p (2.2.7), therefore Vc rank lg £ m-n. 7 l± C Example: Vg = C(t ,t ,t )/tfk ) is not an ideal theoretic complete intersection. (S is the semigroup generated by {3,4,5}). In fact, lg' is generated by {8,9,10}. We will show lqter, that the condition given in 2.2.9. is also sufficient for numerical semigroups of rank 3. CHAPTER 111 Sylvester-semi groups 3.1. Properties of Sylvester-semigroups. For numerical semigroups we do not have to require that they are finitely generated, because this property is common to all of them. In this section we will assume that the generators of a numerical semigroup S have greatest common divisor 1. Because if S is generated by {nj,n 2 , . . . ,n^} and gcd (n j,...,n ^ ) = d, then we obtain a numerical semigroup S' which is isomorphic to S and whose greatest common divisor of its generators is 1. We let S' be the semi group:'gene rated by { Hi ,... . d d It is obvious that if S is such a numerical semigroup then there exists seS such that s+veS for all v=0,l,2,... . Hence for a numerical semigroup S there exists a greatest integer m not belonging to S. In special cases we can compute m, according to the following theorem. Theorem 3.1.1: LetCj,*..,c^eN such that gcd(c.,Cj) = 1 for all i ,j = 1,... ,jG, i t j . Let n. = ff c. for i = 1,2 ...... &. Then j =i J j^i the semigroup S generated by {n^, . . . .n^Jhas the following property: There exists an element mcZ such that for all zcZ we have: zeS if and only m-z i S. & ^ 8l Moreover: m = (jM) ff c. - S iff c. . j=l J i=l j =1 J jVi 20 21 Remark: From the definition of m it follows that m is uniquely determined. Namely m is the greatest integer not belonging to S. Proof of the Theorem: X a.) We first show that if we have an equation E r.n. =0, r.cZ • ill I 1=1 then r. is a multiple of c. for i = 1 A. A A A a Proof: S r. ff c. =E r.n. = 0. Therefore, rt ff c. =0 mod c. -— 1=1 1 j=i j 1=1 1 1 'j- i j 1 j*i .£ jVi for all i = 1 A. But ffc.£ 0 mod c., since the c.'s are j = lJ 1 1 }*\ assumed to be mutually prime. Hence r. = 0 mod c. for all i i = 1,...,X. This is what we claimed. AAA b.)Let m = (X-1) if c. - E ff c. . j=l J i=l j=l J j^i We show that m has the required properties stated in the theorem. 1.) Assume meS then there exist rj,...,r^€Ng such that A m = E r . n. (*). 1=1 A A A- 1 A Nowm = (X-.l) ff. c ...... t,E n...= JD c .n ,,- E n. and therefore (*) j=l J i=l 1 i=i 1 1 j=i ' yields E (r.+l-c.)n. + (r..+J)n. = 0 . Since r.+l > 0 , there exists • i I II JO’ , JO so 1=1 i*{l,... ,X-1 } such that r.+l-c. <0. By a.) r.+l-c. must be divisible by c.. Therefore r.+l is divisible by c.. But 0 < r.+l < c., a contradiction. Therefore nvtfS. 2.) We now prove the converse: If s^S then m-seS. A Let seZ, s^S, s = E r.n ., r.eZ. • I I I i=l\ By the relation c 2 n 2 = we make, in a similar way, the coefficient of n 2 less than c 2 and greater or equal to 0. We can proceed in this way until i = jM . Hence we may assume from the I • beginning that s = 2 r.n. and 0 s; r. < c. for i=l,... ,j£-l. Since i=l siS, r^ must then be less than zero. Therefore jM m-s = E (c.-l-r.)n. + (l-r.)n. is an element of S, since • i I I I Jo Xf 1 = 1 c,H-r. £ 0 and 1-r. s 0. I I Xf • , This theorem gives rise to a general definition. Let S be a cancellative semigroup with 0 element. Definition 2.1.2.: S is a Sylvester-semigroup if there exists me£ such that for all seS*, seS if and only m-s^S. Remarks: We call cancellative semigroups with the above property Sylvester-semigroups, because Sylvester proved Theorem 3.1.1. in the case j&=»2. Some authors cal 1 these semigroups symmetric semigroups; however, the term "symmetric" is also used more commonly to describe semigroups in a different sense. Of course, Theorem 2.1.1 does not determine all possible numerical semigroups, which are Sylvester-semigroups. As an 23 example, consider the semigroup S generated by {4,5,63 . S is a Sylvester-semigroup, but is not of the type as described in 3.1.1. If S is a numerical semigroup then we let M be the maximal ideal M = S - {0} and be the set M «* {zcZ / z+M £ S} . Theorem 3.1.3.: Let S be a numerical semigroup then the following conditions are equivalent: a.) S is a Sylvester-semigroup. b.) M~ = {m} U S, where m is the greatest integer not belonging to S. c.) Each proper principal ideal is irreducible, ( i.e .,: If seS, s / 0 , then the ideal (s) cannot be written as the intersection of two ideals in S, which both properly contain (s)). d.) There exists a proper principal ideal, which is irreducible. Proof: a.) implies b.) : Let zeM . If zeS, then there is nothing to show. Assume z^S. If z £ m, then m-zfM, since S is a Sylvester-semigroup and therefore z + M ^S, since m^S. Thus z must be equal to m. Obviously, meM . b.) implies a.) : Assume S is not symmetric. Then there exists a greatest integer m^S such that m-m^S. We clain: M 2 {m^,m} U S . Proof: We only have to show, that m^eM . Assume m^M , then there exists seM such that m^+s^S. By the definition of mj, it follows that m-m|*seS. Therefore m-mj = (m-mj-s)+s is an element of S, a contradiction. 2k a.) implies c.): Let (s) be a principal idea) and s^eS such that , then s+m e(Sj). Proof: Since s^(s), it follows that Sj-s^S. Therefore m-(Sj-s)eS, since S is a Sylvester-semigroup. In other words: (rtH-s)rSj € S; o or equivalently m fs^s^). c.) follows now easily. Assume (s) is reducible then there exists SpSgcS, Sj ,s^ (s) such that (Sj) a (s2) £ .(s). However, from the considerations above, it follows that s+m e(Sj) /'I (s^) > whereas s+m ^(s), a contradiction. It is triv ial that c.) implies d.) d.) implies a.) : We show, if S is not a Sylvester-semigroup, then each principal ideal is reducible. Claim: If (s) is a proper principal ideal then (s) = (s,s+m)o (s,s+m.j),where m is the greatest integer not belonging to S and m^ the greatest integer not belonging to S such that m-mj 4 S. Remark: By the definition of m and m^, it follows that s+meS and s+m^eS. Also (s,s+m) and (s,s+mj) both contain (s) properly. Thus, once we have proved the claim our assertion follows. Proof of the claim: It is sufficient to show that (s+m) f| (s+m^)s(s). Let s'e(s+m) A (s+nij) then s' = s+m+Sj = s+mj+s2, S j,s 2eS. Assume Sj =0, then m-mjCS, a contradiction. Therefore Sj > 0 and this implies that mfSjCS. Hence s'e(s). 3.2. The semigroup ring of a Sylvester-semigroup, Let S be a numerical semigroup generated by {nj,...,n^} and let k be a field. We know that k[S] =* k[T \ t 2 ,...,T^], T an 25 nl ni indeterminate. LetWbe the maximal ideal (T ) in k[S], K the quotient field of k[S] and Og the local ring kCSj^with maximal ideal= if R is an integral domain with quotient field K and R is an ideal, one defines: A. • = {xeK/x'OtSR} Of course, /Ot- ^ is an R-module and contains R. Hence one can form the R-module A .'/R . The following theorem is of the type as 3.1.3., however on the ring level. Theroem 3.2.1.: The following conditions are equivalent: a.) S is a Sylvester-semigroup. b ) i(aft’/os) = 1 c.) Each proper principal idealin 0g isirreducible, d ) There exists a properprincipal ideal in 0g, whichis irreducible Proof: a .) impl ies b .) 1 .) We firs t show that =^ anc* ^ i sfollows at once as soon as we proyed, that ! = k + k[S], where /Lt is the greatest integer not belonging to S . Let x c w 1, then x ■ , f(T) , g(T)ek[T] . 26 We may assume that gcd (f(T), g(T)) * I, We have T 'ck[S], since xc/yv^. Therefore g(T) =* XTV, X€k, 0 s y s because gcd(g(T),f(T)) = 1. Assume that V > 0, then f(T) must be of the form f(T) = x '+ ..., where x'ck, x' ^ 0. Hence we get T ^ V1 ni”^ ■V'/------= * - T + ... is an element of k[S] . But this can XX only happen, if-W “ n] • If S is generated by just one element, 1.e. n^ = 1, then our assertion follows at once. Therefore we may assume, that S is generated by more than one element. In S Vi 1 s“n, • 1 th is case, there exists scS, with s-n.VS. Hence x T = — T~ +. I /v is not an element of k[S] . Thus V must be zero and g(T) is a constant. Therefore W lek[T]. Now let f(T ) € /V ^ and assume that f(T) =XT1+ . . . , X€k and V a positive integer not belonging to S. We show that u must be equal to fi, thereby proving 1.). Assume. then fi-l/> 0 and. fi-veS , since S is a Sylvester- semigroup. Therefore but f(T)T*i"1' = XT^+.. . is riot an element of k[S], a contradiction. On the other hand, if f(T) = xT^+ h(T), h(T)€kCS], then f(T) is certainly an element of 4W ^ . 2.) We want to show that j&(*2RfC 1 /° s) “ I- This follows immediatel from the fact that ^Og and 1 .), because = ^ 1/ k[S] k[^]°S . Let us prove that ■ S =w'0g . It is trivial that ^ W-vvvbg . Conversely, if xe TfiTlT1 then xftflsOg . Therefore x'WOgSOg. ni Hence xT €Og for i =* 1 Z . Therefore there exist t.,rj,r 2 ...... i>| r j €k[S] such that xT ■ — for i => 1,.. ,Z. This means, however, that txc and thus xc/vH^Og . b ) implies c.): This is a general fact: If (Ri ^ is a one dimensjonal local domain, then the conditions b.) and c.) are equivalent. A proof of this theorem is in [2]. It is triv ial that c.) implies d.) d.) implies a.): Assume S is not a Sylvester-semigroup. Let fij be the greatest positive integer such that>fA ^S and. Let (c) be any proper, pr iricipal id e a lin 0g. >We may assume that cfk[S] . By the definition of ./x. and.fip it follows that ii AM Mi crV (c) and cT i ( c ) , Moreover cT^ and cT are elements in Mi n k[S] . We claim that (c) = (c,cT )A (c^T^). This shows that (c) is reducible. M, _ii Proof of the claim: Let a^(c,cT ) A (c,cr ) . There exist a^,a b^,b 2 e 0g such that U a .= a^c + a2cT = bjC + b 2 cT“ . M, Lt Hence a2cT - bgc r = bjC-a^c. Multiplying this equation by the product of the denominators of aj,a^,b^,b2, we may assume that al,a 2 >bi,b 2 are already elements in k[S] .. n, n Let a 2 = x + a2' , xek, a 2 'c(T ,...,T ) and b 2 = x 1 +^2 ', ni n,0 X'ek, b 2 i.c(T ,...,T ). By the definition of n and .jjj , it follow that ag'T^UkCS] and bg'T^ckCs] . Therefore the above equation 28 Mi ii Mi I, yields xcT - x'cT^ = dcf dek[S]. In fact, d » bj-aj-a2'T -b2'r . Mi Hence we obtain in k[T] that XT - x'T^ 83 d. Since dck[S] , it follows that x « x1 83 0. In other words: a2T and bgl^ are elements in k[S] . Therefore £ *a = (bj + b 2T^)c, £ a unit in 0g and bj + b 2T^ek[S]. Hence ae(c). Remarks: If S is a Sylvester-semigroup, then it does not follow in general that each principal ideal in k[S] is irreducible. O O Example: Let S be generated by {2,3}. The ideal (T +T ) is reducible in k[T 2 ,T3]. In fact, Theorem 3.2.1 is related to a general concept: A noetherian local ring (R,w) is said to be a Gorenstein ring if 1.) (Rj'W) is a Cohen Macauley ring. 2.) Each ideal in (R^w) generated by a parameter system is irreducible. As a special case it follows easily that: A one dimensional noetherian local domain is a Gorenstein ring if and only if each proper principal ideal is irreducible. Corollary 3.2.2.: 0 is a Gorenstein ring if and only If S is a Sylvester-semigroup. ft is known, .that a noethenidn local ■;ring (R,W) which is: a.complete intersection is a Gorenstein ring. Corollary 3.2:3: If.a numerical semigroup S.is a complete intersection then S is a Sylvester-semigroup. The converse Is not true, as we show by an example in Chapter IV. However, there Is a stronger result: Serre proved in [ 5 ] the following: Let k be an algebraically closed field, C a curve in the affine 3-space A^(k), PfSC a point of C and k[C]p the local ring of P on C. Then the following conditions are equivalent: a.) k[c]p is a complete intersection b.) k[C]p is a Gorenstein ring. Corollary -3.2.4.: If S is a numerical semigroup of rank 3, then S is a complete intersection if and only if S is a Sylvester- semigroup. In Chapter IV we will prove 3.2.4 directly and give some other equivalent conditions for a numerical semigroup of rank 3 to be a complete intersection. CHAPTER IV Numerical Semigroups of Rank 3 k . i . The minimal relations of numerical semigroups of rank 3. Prei iminarv remarks: Let S be a fin itely generated canceliative Nn semigroup, say S = 0/p . To a relation Aep, A ** (v,v‘) we assign a vector in Zn in the following way: A -* wA = v-v 1 . It is easily verified that the set Mp(S) = {wAezn/Acp} is a subgroup of Zn . We call M (S) the relation module of S (with respect to p). If we P are given M (S) we can recover p. To show this,-wfr• introcjuce some- - P notions: Let v, v'eZ^; v = (z.), v 1 = ( z .1). We ■def ine v v v 1 = (max( 2 . , z . ') ) , v a v ' = (min(z. ,z. 'i)) *4* ■ p .j, M and v = vvO, v = (-v)vO . For any vcZ we have v = v -v We define the set of reduced relations of p to be Pr = {Aep / A=(v,w), vA w = 0} . It is clear that p = pr . In fact p = {(v+w, v'+w) / weZn, (v,v 1)€pr } . Therefore pis completely determined by pr .There is a bijection between the elements of pr and the elements of M (S), which is established in the following way: 30 Hp W t------► (w+,W ) a A, W WA = v-vU ------,(v,v') “ A. The above considerations show how to obtain.p from M (S). There- fore S is completely determined by M (S) . We say that V| cMp(S^ 9enerate P the relations \ = (vj+»v|"). 1 “ generate p. If {A. = (v.,v.')/i=l k] i is a minimal system of generators of p, then the relations A. are reduced, since S is cancel 1 ative. Therefore to find a minimal system of generators of p is equivalent to finding Vj, . . . fV^fM (S), k minimal, such that vj ,... ,v^ generate p . The minimal relations: Let us return to the case where S is a numerical semigroup of rank 3. Say, S is generated by {nj,n2,nj}, n.eN . Let e. = (1,0,0), e , = (0 ,1 ,0 ), e, = (0,0,1). We have an epimorphism < N 3 p* : -♦ S, where p’f(e.) = n. for i = 1,2,3. Hence S = 0 / , p * {(v v'.)/p*(v) = p*(v')} and f ? 3 Mp(S) ={(z1 , 22,z3)cZ A S z.n. ■* 0} . If v € M (S), v ^ 0, v = (zj ,z2,z3), then there is one component of v, P say the i - component, such that either: 1.) z. > 0 and z. £ 0 for j ^ i i j or 2.) Zj < 0 and z. z 0 for j ? i . 32 We say in this case, that v is of type i and define: ------Definition 4.1.1: v € M p(S) is a mimimal relation of type i if for al 1 relations v'eM (S) of type i, v* » (z‘), we have P J h ' N * , ! • A relation is said to be minimal if it is minimal of any type. We denote the set of minimal relations of M (S) by Remark: The minimal relations of type 1 are obtained in the following way: Take the smallest multiple Cj of n^ such there exist r2, r^c NQ with c^n 1 = r^ ^ r ^ . Then v = (-Cj,r 2 ,rj) is a minimal relation of type 1. Similarly one finds the minimal relations of type 2 and type 3. From this description follows immediately that ?RfLIs finite. Let Cvj ,v2, ... ,v^} £ Mp(S) be any minimal set of generators of p. We will prove later that {V| ,v 2 v^JS This shows that it is worthwhile to study the minimal relations of M (S). Let us choose Vj ,v2>Vj e $fl/, V 1 = ("Cl ,r 12,r 13^ v 2 “ ^r2 1 ’ " C2 ’ r 2 3 * * Ci> ° f° r ' " 1 , 2 , 3 V3 = ^ 3 1 ’ P32’ " V and rIJi 0 f°r j =1»2,3 ‘ We will distinguish two cases: Case I: r.j 0 for all i,j = 1,2,3. Case II: There exists i,je{l,2,3} such that r ^ = 0. 33 First we consider Case I: Proposition 4.1.3.: I n case I we have, v = v, + v 2 + v3 - 0 . Proof: Assume v ji 0 and say v is of type 1. Ifthe firs t component of v is less than zero, then -c^ + r 2j + r-jj < 0 . However, since c^ is minimal for relations of type 1, we have that ” C 1 + r 21 + r 31 S ”dl ’ a contradiction, since r2j > 0 and r ^ > 0 . If the first component of v is positive, then -c 1 + r 2 l + r3l > ° ’ And again, since c^ is minimal we have -c^ + r2j + r ^ ^ c^. But then either ^ c^ or r ^ S: c^ . Say r2] ^ c^, then V 1 + v 2 = ( r 21 "cr r 1 2 " C2 ’ r l 3 + r23^’ Where r 2 i ' C | a 0 and r 13 + r 2 3 > *"*ence r i 2 " C2 ^ ® and therefore v^ + v 2 is a relation of type 2, But c is minimal for relations of type 2, therefore r ^ 2 - c2 s: -c2, or equivalently r ^ 2 £ 0. This is again a contradiction, since r ^ 2 > 0 by assumption. The proof works similarly, if we assume that v is of type 2 or type 3 . i Proposition 4.1.3.: In case I we have TlJl “ C± Vj, ± v2, ± v3) . Proof: Let v1' = (-c^ , r 12‘ * *" 1 3 ') be an element offfiC. If °» then Vj'-Vj = (0, r ] 2 '"rl2,-r13^ is a re,ation of tVPe 3- Therefore Tjj i Cj. We know, however, by 4.1.2 that r ^ a c 3 ~r 23 and hence- r._ < c,, since r9« > 0, a contradiction. Similarly it can be shown I j J 34 that r]2' t 0. Thus we may restrict our attention to the case where v ] 1 53 (-Cj »r^2 ' *rl 3 *^ w*th r 12* > 0 and r^ ' >0. We appiy again 4.1.2. and obtain that V|'+v 2+v3 = 0. Therefore Vj'“ -(v 2+v^) = Vj. Similarly it is shown that v 2 and are uniquely determined. This proves 4 . 1 . 3 . Next we consider case II: Proposition 4.1.4: In case II either a . ) (0 ,-c 2 ,c3) € Tfttl or ...... £•) (c,,0, -c3) etfJlor y.) (-C, ,c 2 ,0) € '$ ft Proof: We may assume without loss of generality that r2j = 0. Therefore, v 2 = (0, ~c 2 *r 2^ ) • 1f r23 = °3 ’ then °Ur assertIon follows. Thus we may assume that r 2 3 > c^, since is minimal. Consider v 2 + r32 "°2’ r23~C3^* s *nce r 3 i ^ 0 and r 2 3 -c 3 > 0, we get : r 3 2 -c3 < 0. This implies, that r 3 2 »c2 £ -c2, hence r 3 2 £ 0. However, r3 2 it 0, therefore r 3 2 = 0 and r ^ £ Cj. If = Cj, then our assertion follows again. Thus, let us assume that r ^ > Cj. Now Vj + = ( r 3 i“c i» r 12*r 13 "°3^ iS a relation of type 3, since r 3 j”c] > 0 and r \ 2 a Therefore Tj3 -c3 < 0. By the minimality of this means that £ -c3, hence r j 3 = 0. Thus Vj ® (-C |,r^ 2 ,0) and r j 2 £ c2> This is, however, a contradiction, since vl+v2+v3 " (r3l’cI ,r!2"°2,r23’c3^ *s/a re,ation with r 3 l"cl> °» r ] 2 -c 2 £ 0 and > 0. Therefore r3j = Cj and v 3 « (Cj,0 -c^) which proves our proposition. In the following proposition we describe explicitely the set of minimal relations in case 11 . If x is a real number, we define [x] to be the greatest integer less than or equal to x. Proposition 4.1.5.: In case II the set of minimal relations is either: {±(Q,-c2 ,c3), ± (d(0 ,-c 2 ,c3) + (-c,,rl 2 , r I3))}, r ■ r where deS and -[■— ] s. d s [ ~ ] , r 12> r J 3 :s 0 . or I s3 ± (d (c p 0 ,-c3) + (r2i *“c 2 » r23))}» r r where dcZ and - [-—4 i d s r21 * r23 & °* or y.)$St= (± (-CpCg.O), ± (d(-c 1 ,c 2 ,0 ) + ( r ^ ,r 3 2 ,-c3) )} , r r where deZ and - [ - | 4 £ d :£ C ^ -], ^ , r3 2 * 0. Proof: Referring to 4.1.4 we may assume that (0,-c 2 ,c3 )eD$T. Let v] = »ri2 ,r ]3, ) , v2 - (0,-c 2 ,c3) and^=.{± Vj , ±(dv 2+Vj)}, r r - [—^] £ d £ [—^ ] . We show that «■ 7f[/. °3 c2 It is clear th a t^ s T ^ • Conversely, assume we are given a minimal relation v2' - (r2J1,"c2*r23'^ o f type Z‘ 1f r23' = ° 3 » then v2 '-y 2 = (r 2 ^',0,0). therefore r 2 j ' = 0 . In other words v2' = v 2 and thus v2' e/Jl, . If r^ ' / C y then v 2 '-v 2 => ( r 2 1 1 ,0 ,r 2 3 '-c 3) is a minimal relation of type 3. Since r 2 j 1 2; 0 we have r ^ '- C j < 0. 36 Then r^'-c^ -cy because of the minimality of c^. So that in this case r ^ ' . 83 °* Hence v2‘ - ( r 2 1 ' ,-c 2 ,0 ). Now Vj+v2‘ ■ (r ^ 1 *ri2“’c2*r l3) is a relation of type 2, since r 2 i'"ci ^ 0 and r^ it 0. Therefore r l2~°2 ^ °* 1 f r 12~c2 * then r2 i' ° C1 * an<* r 13 “ °* Hence v2‘ = v 1 and thus v 2 * c ^ . If r 1 2 “ c 2 < °* then r 1 2 ~ c 2 ^ “c2 * But this implies that r ^ 2 = 0, so that v^ » (-c ^ ,0 ,rj^ ). Now vi+v 2 ' " v 2 = (r 2 i ' ”ci*°»r13”°3^ is a relation wJth r 2 i ‘ “ci ^ 0 and r l3”c3 ^ °* Hence r 2 j 1 = Cj and r ^ = c^ and v2' = v 2 -v i • In any case we get that v 2 'clf[/. Similarly it is shown that any minimal relation Vj 1 of type 3 is an element ofltf/. I f Vj 1 = (-Cj , r 1 2 ' , r-1 3 ') is any minimal relation of type 1 ,- then Vj'-Vj = (0 , r i 2 , - r i 2 *r 1 3 ' ~r 1 3 ) m dv2 " where deZ< The s 'de conditions for d are obvious. Hence Vj 1 = vj + d v 2 and our proposition follows. Our next aim is, to prove that the minimal relations generate p. Let R be a commutative ring with 1. If v f H (S), we define H Fy c RCXp^.Xj] to be Fy = XV+- XV~. As an immediate consequence of 2 . 1 . 5 . we obtain, ^ r [ x , , x 2 , x J RCs] = ' 2 3 / ({F } ) • vvfMp(s) la order to show that the minimal relations generate p we only have to prove, according to 2 . 1 . 6 ., that ([ F V) V€^ ) “ ^ Fv^vcM (S)^ ^ * P We define on S an ordering : if s^s^S , then s^ Ss s 2 if s^-s^S and we say that Sj > s 2 if Sj it s2 and s j ^ s 2 • It Ts clear that: 1.) If Sj,s 2cS, then s ] ^ s2, s ] t If and only If s } ■ s2. 2.) If £s.}i=1 2 S S such that s |+I * s. for all I » 1 ,2 ,..., then there exists i^ such that for all I ^ 1^, s.+j * Sj . In Chapter II we introduced on R[XpX 2 ,X^] on S-graduation. If fcR[x,,x 2 ,x3] is a homogeneous polynomial, then we denote its homogeneous degree by deg (F) and for two homogeneous polynomials F, 6 , we mean by deg (F) < deg(G), that deg (F) is less than deg (G) with respect to the ordering on S. Now (*) will be a simple consequence of the following proposition. Propos it ion . 1. 6 . Let v e M^(S). Then Fy can be written as Fv = F' + aFw’ where F' e ^ Fv}vc'fll) ’ w e V S)’ Qc R^X 1 »X2 »X3 ] and deg ( F j < deg (Fv). Proof: Case I: 2$ l= {±Vj, ±v2, = ^ 3 , where (“c i »r]2 ,r 13^ ’ v 2 = ( r 2 i ’ " c 2 >r 23^ ’ v 3 = ^r 3 I , r 3 2 , - c 3 ^ a n d f o r a1 -1 ' 1»2’3’ r -IJ • ^ > n 0. Let v e M (S). If v e l $ l , then the assertion is trivial. Other- P wise we may assume without loss of generality that a,-c, v = (-a r a 2 ,a3), a| > c , , a2 i 0, a3 2: 0. We get Fy - X, F « mi n(a 2 ' r 1 2) min(a 3 ,r 13) X2 . X3 . F , where a2-min(a2,rJ2) a3-min(a3 , r 13) aj-Cj r J2-min(a2,r J2) Fw = X2 X3 " X1 2 r -min(a r. ) X3 . Of course w e Mp(s) and 38 a.-c. min(a r._) mln(av r , J deg (Fw) - deg (F^X, '-F ) - deg (X2 ,Z -Xj 3 13 ) min(a r. ) m in(a,,r.-) - deg (Fv) - deg (X2 12 . Xj 3 13 ) . It cannot be that (a 2 ,a^) = (0,0). Therefore min(a 9 ,r19) min(a_,r.-) deg (X2 2 12 • X3 3 13 ) > 0. Hence deg ( F j < deg (Fv). This proves our assertion in case I. Case II: We may restrict our attention to considering only II a.), *®= C± v2, ± (d v^Vj)}, v, - (-c,»r| 2 ' r 1 3 ^' v 2 “ (°»"c2 ' c3 ^ and - [ I n j * d s [~“ ] . 3 2 1.) Let v = (-a j,a 2 ,a^) be a relation of type 1. Hence a ^ Cj , and a 2 S: 0, a^ s 0. If a 1 = Cj then our assertion is trivial. Therefore we may assume that a^ > c^. We get similarly as in Case I a.-c. min(a,,r.,) mln(a,,r._) Fy - x , 1 ' Fy = X2 2 ■ X3 3 '•> • Fw with w { Hp(s). If a2 > 0 and| a^ > 0 then deg Fw < deg Fv and the proposition follows Now assume that a2 = 0. If r ^ > 0, then again deg Fw < deg Fy, since a^ > 0. I f r ^ = 0, then r J2 £ c2 and Vj' = vj +v2 = ( -c .,r.- -c .,c«) is a minimal of type 1, whose third component is at-ct min(ao,c,) greater than zero. Therefore Fy - Xj Fy ,= Xj • Fw, , w‘ e Mp(S) and deg Fw,- < deg Fv> since min (a^,Cj) > 0. The proof works similarly if we assume that a^ = 0. 39 2.) Let v = (aj,-a 2 ,aj) be a relation of type 2. We may assume that a 2 > c2, otherwise there is nothing to prove. If a^ = 0 , then v is also a relation of type I and this case is already a 2 - c 2 min(a,,c*) treated in 1. If > 0, then Fy - X£ c L• Fv " x 3 * Fw» where w c M^(S) and deg Fw < deg Fy, since min(a^,c^) > 0, The case that v e M (S) is a relation of type 3 is similarly proved P as the case above. This completes the proof. As a corollary of 4.1.6 we get : Theorem 4.1.7: The minimal relations of M (S) generate p. Theorem 4.1. 8 .: In case I : S is not a complete intersection. In case 11:: Sis a complete intersection. Proof: Case I : I = (F ,F ,F ), where b v, v 2 v3 ’’j 2 r 13 C1 Fv, = X2 X3 ' V r 21 r23 ° 2 Fv2 ■ X 1 x3 - 2 F - X,3' X2 3 2 - Xj3 and r,j > 0 for i ,j = 1,2,3. Assume we have an equation Fv, = Fv 2 + «3 Fv3 ’ 1 ■ ’ •2’3- This equation yields ci - **21 r9? - C3 -Q.J Xj =■ a 2 X, X3 * - Q3 X ^ mod X2> where (ij - Qj mod X2, i = 1,2,3. Since r2j > 0 we have Qj® • P with PfRtXj.X^ . On the other hand: ^ ® Q, - X 2 • T with TfR[X,,X 2 ,X3] . Hence 0, - XgT + X3P . Therefore Qj c(X],X 2 »X3). The Ideal lg Is contained In (XpX 2 ,X3). Let^-be a prime Ideal In R[Xj ,X 2 ,X3] containing (XpX 2 ,X3), then Qj Is a non-unit in the local ring R[Xj ,X 2 ,X3] ^ . Similarly, starting with equations Q 2 Fv = QjF^ + “ ^ lFVj+ ^2 Fv3 leads to the conclusion that are non-units in R[Xj,X 2 ,X3] Therefore rank (lg RCXpX^Xj]^ ) = 3 and S cannot be a complete intersection. Case II: We may assume that 'fttft® {±(0 , - c2 ,c3), ± (d(0 ,-c 2 ,c3) + (-c| ,r12»r13))} w'th r r deZ andd s: [-11] . Let v = (0, -c 2 ,c«) and c3 2 r r V 1 = ^’cl ’ rl 2 + *-"c^c2’ r 13 " C“^ ] c 3? " ^"c l ,r 12, r 13^ * Then for any vc Z f y w e have either v = ± v 2 or v = ifdv^+Vj), r r d ® 0,1,2...... [ - ^ ] + [ - ^ ] . We claim: I = (F ,F ) . 3 2 * V1 2 Proof: Let ve'flfl/, v = dv 2 +Vj, d > 0, then v - (-Cj ,r^ 2 -dc2 , r 13+dc3) r . 2 “dc2 r l 3+ ^ c,5 C 1 and Fy “ x 2 X3 xi • Obviously there exists a polynomial Q,cR[Xj ,X 2 ,X3] such that Fjv 83 Q.Fv . ■ X2” 2 ^ - Xj1 a Fv. This proves the claim and completes the proof. Corollary k . 1.9.: In any case rank p s 3. In the following section we give some equivalent conditions for a numerical semigroup to be a complete intersection. k . 2 . Equivalent conditions for complete intersections. Let S be a numerical semigroup of rank 3 generated by ft 3 {nj,n 2 ,n3}, n.e N and let p : Njjj -* S be the canonical epimorphism, defined as in 4.1. We define Ig* = {seS/ There exist such that piM ') = p(4^)= s} I lc Is an Ideal In S. In fact I. = ( td e g Fv}vcM (S) ) . For v eiys) Theorem 4.2.1.: The following conditions are equivalent: a.) S is a complete intersection. ft b.) Ic is generated by 2 elements. c.) p i s . d.) S is a Syivester-semigroup. 42 Proof: We have proved already that a.) Implies b.), see 2.2.9. b.) Implies a.)j If S Is not a complete Intersection then we know that ■ -£± V|, ± v2, ± Vj3 with V1 “ ^“c r r12* rl 3^ v 2 * ^r2 1 ’ ~c2 ’ r23} v3 “ (r3 1* r3 2 » “V and rij > 0 forl,J = ,»2»3* lsIs certainly generated by {s(vj), s(v2 ), s(Vj)}. We only have to show that for i , j = 1 ,2 , 3 , i • j* j, s(v.)-s(vj) & S. We may assume that } = 2 and j = 1. The other cases are proved similar. Assume s(v2) " s(Vj)eS. Then there exist a^a^a^Ng such that c2 n 2 ’ °lnl = alnl + a 2 n 2 + W Hence ^c 2 "a 2 ^ n 2 + ^”° r a l^nl " a 3 n 3 = °* Since "aj ^ 0 and “c| “a i < °» we have c 2 -a 2 > 0 . This implies that c^a,, ^ c2< Therefore a^ = 0 . Thus we obtain the relation v a (-c^“ap c2 »a3 ) > which is minimal of type 2 . Therefore v = -v2. Hence r 21 » a f + c] i Cj. However r21 = c]“r 3 i < c]» since r 31 »* 0, a contradiction, a.) implies c.) : We may assume that we have the minimal relations vl - ^”c l »rl2 *r l 3 ^ and v 2 = (°»"c2 »c3 ^ * Obviously, ju = c^nj + c 2 n 2 - n ^ n ^ n ^ ® c|n| Cjiij nl"n *** 2 * n 3 * We have to show that M * S. Assume there exist a ^ a ^ a ^eNg, such that ft = a ^ + a2 n 2 + a^n^ . Then (aj + l-Cj)nj + (a 2+l)n 2 + (a^+l-c^n^O. 43 Using relation v 2 we may assume that 0 s a^ < c y Hence a2+l > 0 and a^+l-Cj £ 0 and therefore a^+l-c^ < 0. This implies that ai+l-C| & -Cj .Therefore a^ < 0, a contradiction. c.) implies a.) : If S is not a complete intersection then ' $ 1 = {±vr ±v2, ±v3) with V 1 " ^-CJ’ r 1 2 * r 13^ v 2 = ^**21 ’ ”C2 ’ r 2 3 ^ V3 = ^r31’ r32* -c3) and r.^ > 0 for i,j » 1,2,3. We want to show thatc S. Without loss of generality we mayassume that s(vj) < s(v2)< s(v3). Then H = s(vj) + s(v2) -n] " n 2 " n 3 = + c 2 n 2 - nr n 2 " n 3 “ r 12n2 + r 13n3 + r21nl+r23n3 " V n2"n3 = (r2 j-l)nj + (r12“ 1) n2 + (c3 -1)n3 . The last equality follows from v, + v2 + Vj = 0. Since r2j > 0, r.j2 > 0 and Cj > 0, / 1 ( S. a.) implies d.) : We may assume that we have the minimal relations Vj = (-Cp r j 2, r j 3) and v 2 = (0, -c 2 ,c3). In order to show that S is a Sylvester-semigroup we prove: If z e Z, then z f S if and only M"z V S. One direction is clear, since we know already that if S is a complete intersection then H £ S. To prove the converse, assume z € Z andju-z £ S, We want to show that z e S. Assume z £ S, z =» a j ^ + a^nj + l*3 n 3 » a j*z* Using the relations Vj,v 2 we may assume that 0 s: a^ < c^ and 0 s: a2 < c2 and therefore < 0 , since z i S. Therefore fj,-z = (-a^-IJn^ + (Cj-l-a^nj + (cj-l-a 2 )n 2 and -a^ -1^0, Cj-1-a^ i 0 and c 2~^~a 2 ^ ®* hence fi-z e S, a contradiction. d.) implies a.) : Assume S is not a complete intersection. Then {± V|, ± v2, * v3 } V 1 “ ( " V r 1 2 » r1 3 ^ v 2 " ^r 2 1 ’ "C2 ’ r23^ V3 = ^r31* r 3 2 * "c3^ and r ij > 0 for = We c,ain that S is not a Sylvester-semigroup. Let us first prove a simple lemma. If S is a Sylvester-semigroup (numerical) and z e Z, z £ S, then there exists s f S, s £ 0 , such that z+s £ S, or z is the greatest integer not belonging to S. Proof of the Lemma: Let m be the greatest integer not belonging to S. We may assume that z ^ m,therefore z Proof of the claim: Let jl, = Cjn, + c 2 n 2 - n 1 "n 2 “n3“r i2n2 and ^2 = c lnl + c2n2“nl“n2’n3‘ r21 Claim 1: p . £ S, i = 1,2. Proof: Assume ^ e S. Then there exist a ^ a ^ a ^ eNQ such that P>) - a in) + a 2n 2 + a 3 n 3 NoW * ^ 1 “ Cl n 1 + c2 n 2 "nr W r 1 2 n 2 = (c^-1)n 1 + (r^-O ^-n^. For the last equality we used Vj+v 2+Vj => 0. We may assume that 0 ^ a^ < c^ and it follows that Assume c^-l-a^ £ 0, then rj2"^"a2 > *mP^es t*iat r 3 2 "l “ a 2 25 c2 * since r j 2 + r32 " c 2 We 96t " , - a 2 * r 12 > °» hence a 2 < 0 , a contradiction. Assume r ^2~^~a2 * °» tlien c l“ ^“al > 0 and theref°re c^-l-aj 2: Cp hence a^ < 0 , a contradiction. We have, therefore, that c^-l-a^ > 0 and r32~^’a2 > ® hence -l-a^ ^ -c^. it cannot be that l-a^ < -c^, since by assumption 0saj< c^. On the other hand, assuming that " 1 " a 3 * cj» then (cj-l-a,, r 3 2 -l-a2, -l-a3) = v^ Therefore r 3 2 -l-a 2 = r 3 2 . Hence = -1, a contradiction. In the same way it is shown that M2 i S. Claim 2: (l. + seS for ail seS, s # 0, i=1,2. Proof: We show this only for H y Similar arguments work for ,^2. It is enough to prove that jUj + njcS for i = 1 , 2 , 3 . ^1 + nl “ Clnl $ v2 n 2 ’n 2 “n 3 “r 1 2 n 2 = (c2 -l)n 2 + ( r 1 3 " 1) n 3 Hence jij + n^eS, since c2»l £ 0 and r^-1 s 0, Wl + n 2 * c lnl + c2n2_nr n3“r 12n2 = (r 2 i“1)ni + (c3-,)n 3 Hence,/ij + n 2 eS, since r 2 j-l is 0 and Cj-l ^ 0. + n3 - cini + c2V nl"Vr12n2 (Cj“I)nj + (r 3 2 “, )n 2 . Hence + n3eS since Cj -1 ^ 0 and r32-l £ 0 . 46 Now assume- that S Is a Sylvester-semigroup and m the greatest integer not belonging to S. We apply our lemma and it follows that fij » m and /i2 ** m. Therefore, =» ju 2 and, hence, r 2 1 nl •* *"i 2 * ^ 2 *s a contradiction, since 0 £ r J 2 < c2 , Coroilary 4 .2 .2 .: If S is a numerical semigroup generated by £n,,n 2 ,n3J and S is a Sylvester-semigroup then jU ■ min {s(v1) + s^v^J^nj-n^n^ is the greatest w V s) v ^ zy2, zeZ integer not belonging to S. Conjecture: Referring to the proof of 4.2.1. the author conjectures that max.(Hj ,ji2) is the greatest integer not belonging to S. For tirs sake of completeness we show by examples that most of the equivalent statements of Theorem 4,2.1 are no longer equivalent if the rank of the semigroup is greater than 3 . Example A: Let S be the semigroup generated by £5, 6 ,7,8 }. S is a Sylvester-semigroup but is not a complete intersection, beicause Ig is generated by £12,13,14,15) and hence rank Ig = 4. However, rank Ig s hould be 3 . A reasonable definition of the number ./i for numerical semigroups S of rank 4, generated by Cn| ,n 2 ,n^•n^}, would be ,,/u =* min £s(Vj) + s(v2) + s(Vj)}-nj-n 2 -nj-n^ . V W Mp(s) vl ,v 2 ’ v 3 Hnear,y *nt*ePent*ent* h i If we take this as definition of /i, then In our case, fi = 13, hence fie S. Therefore, if S is a Sylvester-semigroup then it does not follow in general that jU^S. Example B: Let S be the numerical semigroup generated by {6,8,9,10}. In this case rank l§ = 3. In fact lg is generated by {16,18,20}. We show, however, that S is not a complete intersection. We have relations v1 = (I, -2, 1, 0) , s(V|) = 16 v2 = (0, 1, - 2 , 1) , s(v2) = 18 v3 = (- 3 , 1 , 0 , 1 ) , s(v3) = 18 = (- 3 , 0, 2 , 0) , s(V/f) = 18 v5 = ( 2 , 1 , 0 , - 2 ) , s(v5) = 2 0 and so on. If an<* v j 6 v. for i = 1,2,3,^,5 then s(v) > 20. r ' We see that F = F - F y k v3 V1 Assume F «■ d } Fv + QgF , Q. e R[X, .X^X^.X^] then -Xj 3 = 0 mod.(X9 ,X«,X:,.), a contradiction'. Assume F = Q..F + Q.,F , fc J T 2 2 3 Q. eRCXi^ ,X2 ,X3 ,X;^] then -X .^2 b 0 mod (X.'j ,X2 ,X:^), a contradiction. . Therefore, if S were a complete intersection then Ig would have to be generated by {F , F , F } . But F £ (F , F , F ), since V 1 2 3 5 1 2 '3 s(Vg) - s(v.) *S for i = 1,2,3. U8 Quest ion: Let S be a numerical semigroup generated by {npn 2 ,n^f. .. tn£} and define (i = min {s(vj) + .. " ni ~n2 ~ ’ ' * "rV vi VicVs) i linearly independent. I jB- I Are the following conditions equivalent? 1.) S is a complete intersection. 2.) fi i S. BIBLIOGRAPHY 1. R. Ap^ry, Sur les Branches SuperIin£aires des Courbps Algebrlques, C. R. Acad. Scl. Parts 222 (19W), 1198-1200. 2. R, Berger, Ueber etne Klasse Unvergabelter Lokaler Rlnge, Math. Ann. 1^6(1962), 98-102. 3. A. H. Clifford and G, B. Preston, The Algebraic Theory o f Semigroups, Vol. II, Am. Math. Soc. (1967), 125-136. k . L. R^del, The Theory of Finitely Generated Commutative Semigroups, Pergamon Press. 5. T. P. Serre, Sur les Modules P rojectifs, Sera. Dubrell- Pissot (1960/61), Exp. 2. ^ 9 BIOGRAPHY The author was born on December 21, 1941, in Heidelberg, Germany. In 1963 he began graduate study at the Universitaet Kiel and in May 1964 he transferred to the Universitaet Heidelberg where he received his Vordiplom. The following year he entered Purdue University and received a Master of Science in Mathematics in June 1968. In September 1968 he continued his graduate work at Louisiana State University where he is presently a candidate for the degree of Doctor of Philosophy in Mathematics. 50 EXAMINATION AND THESIS REPORT Candidate: Juergen Gerhard Reinhard Herzog Major Field: Mathematics Title of Thesis: GENERATORS AND RELATIONS OP ABELIAN SEMIGROUPS AND SEMIGROUP RINGS Approved: £ . K o / u f M ajor Professor and C hairm an D ean of the G raduate School EXAMINING COMMITTEE: ( I t , 4 . Y r d L . ^ O L * * t ______ Date of Examination: June 12, 1969