Regular Sequences and Resultants Research Notes in Mathematics

Volume 8 Regular Sequences and Resultants

Giinter Scheja Universitdt Tubingen Tubingen, Germany Uwe Storch Ruhr- Universitdt Bochum Bochum, Germany

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Library of Congress Cataloging-in-Publication Data

Scheja, Giinter, 1932- Regular sequences and resultants / Giinter Scheja, Uwe Storch. p. cm. -- (Research notes in mathematics ; v. 8) Includes bibliographical references and index. ISBN 1-56881-151-9 1. Elimination. 2. Projective spaces. 3. Intersection theory. 4. Sequences (Mathematics) I. Storch, Uwe. II. Title. III. Research notes in mathematics (Boston, Mass.); 8. QA192 .832 2001 512.9'434-dc21 2001022483

Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the Psychology Press Web site at http://www.psypress.com Preface

In these notes we develop aspects of elimination by treating regular sequences and resultants. We consider quite generally anisotropic projective spaces which gained increasing attention in the last decades. Arbitrary noetherian rings will be used as base rings, and the representation runs throughout in terms of commutative algebra, which has the positive side effect that the pattern of specialization appears fitting. The key concepts are regular sequences and complete intersections. Their use benefits by the well-developed and clear duality theory. A reading of these notes thus calls for a basic knowledge of commutative algebra and of homological methods, the features of Koszul complexes, for example. We made some effort, though, to give complete and down-to-earth proofs for those parts which depend on special constructions. More details are supplied in the following descriptions of the individual chapters. Chapter I deals with two topics which do not depend on the material of the other chapters and hence can be read independently. Sections 1 and 2 treat Kronecker's method of indeterminates. The concept of Kronecker extensions of a ring and its modules is introduced and developed in a general set-up. These sections also provide typical examples of the kind of methods of commutative algebra used in the notes. The second topic is the study of numerical monoids, i. e. submonoids of the additive monoid N of non-negative integers containing almost all elements of N. Such a monoid turns up in a natural way as the monoid Mon(7) = Mon(7o,..., 7n) generated by the (positive) weights 70,..., 7ra of the indeterminates TO, ..., Tn in a polynomial algebra A[T] = .A[To,..., Tn] which defines the anisotropic projective space IP7(y4.) = Proj A[T] over the commutative ring A. At first reading it suffices to look through Section 3. Chapter II contains the treatment of two of the fundamental concepts which we will use extensively in the later chapters, namely the concept of regular sequences and the concepts of (relative) complete intersections and locally complete intersections. This chapter may also serve as an independent introduction into methods which have useful applications in algebraic geometry as well as in complex-analytic geometry. Basic constructions and lemmata are treated in Section 6, whereas Sec- tion 7 features specialities of the graded case.

V VI Preface

Crucial for what follows is the treatment of — as we call them — generic polynomials in Section 8. Generic polynomials are (not necessarily homo- V geneous) polynomials Xli/ejv UVT G Q[T] = Qpo,..., Tn], where the set N of tuples of exponents v = (i/o, • • • > vn) is a finite subset of Nn+1\{0} and where the Uv are (different) indeterminates in some polynomial ring Q over Z. The fact that a sequence F$,... ,Fr of such polynomials is regular, and that the Lasker-Noether decomposition of the ideal they generate has in addition suitable properties, can be characterized by combinatorial means. This leads to the concepts of admissible arid strictly admissible sequences of generic polynomials, which may well be of interest in other parts of algebra. As a special example we treat sequences of generic binomials. In Section 9 we prove a combinatorial characterization of algebraic in- dependence of generic Laurent polynomials in Q[T, T~1]. This will be used frequently in what follows. Chapter III presents the main case of elimination (with respect to projective spaces). In Section 10 the necessary prerequisites are briefly developed. The elimination ideal TO , part of the ideal of inertia forms T, is introduced, and the so-called main theorem of elimination is proved. In Section 11 we switch to the main case of elimination, which centres around the case of n + 1 homogeneous polynomials FQ, ..., Fn in n + 1 indeterminates TO, ..., Tn . In a first step generic sequences are treated. For many purposes it suffices to handle regular sequences of generic polynomials, i. e. admissible ones. In the strictly admissible case more satisfactory results are derived, including the computation of the degree of elimination. Section 12 is a substantial part of the notes. The ideal of inertia forms and the elimination ideal with respect to an arbitrary regular sequence of homogeneous polynomials FQ, ..., Fn G -A [To, • • • > Tn] are determined. The means for this are provided by an extended version of duality for graded complete intersections, which is developed and discussed in detail. Thereby the ideal of elimination is shown to be divisorial if A is an integrally closed noetherian domain, hence a principal ideal ^ 0 if A is even factorial. Chapter IV, finally, deals with the resultants. To begin with we introduce in Section 13 the resultant ideal 91 = 9l(.Fo,..., Fn) for a regular sequence FQ, ..., Fn of homogeneous polynomials of positive degrees in the polynomial algebra .A[To,..., Tn] over an integrally closed noetherian domain A. The ideal 9t in A is a divisorial ideal like the elimination ideal TO of the polynomials and has the same zero set as TO , but it has better functorial properties and is always principal (which is not true for TO in general). Also it suits convincingly the geometric situation: The degree of 9t defines the degree of elimination in a proper way. This is the main result of Section 14 Preface VII and is again an application of straight duality theory for regular sequences. In addition, these methods yield further characterizations of the resultant divisor. In Section 15 we use the Koszul resolution to construct a canonical 7 generator R = R(J b,..., Fn) of the resultant ideal 9l(.Fo,..., Fn), which we call the resultant. Going back to the generic situation one can define the resultant for an arbitrary sequence FQ ,..., Fn of homogeneous polynomials in -4[To,..., Tn] over any base ring A in a unique way up to sign. In the classical case with weights 70 = • • • = 7« = 1 the element R & A is the classical resultant already known in the 19th century and described, for instance, by A. Hurwitz in [19]. In the last section we present some properties of resultants which are valuable for calculations, for example the norm formula and the product formula. Supplements are added to every section. Some simply provide exercises fitting the context or supply further information and examples, for instance showing that inequalities are sharp or that certain hypotheses are indis- pensable. Some describe details of tools or results generally known in commutative algebra, but not readily available in the literature. And some add material to special points we like to emphasize but couldn't pursue in the main lines of development. Added hints to crucial arguments and other remarks, put in brackets, are sometimes elaborate, at least in those supplements serving the main text directly, where the reader should not be left alone. The comments are also detailed in supplements which follow a theme of their own and in which the reader might be interested for various reasons. In general, we have taken care that it is unnecessary for the reader to go through the supplements in order to be able to follow the main ideas. In addition, those supplements deepening sidelines or following historical paths or those which may finally be of interest to specialists only, are given the sign f at the beginning as a warning. These supplements can safely be skipped in a first reading. Prerequisites on commutative algebra are best taken from Atiyah/Mac- donald [1], Serre [45], Matsumura [31] and then Bourbaki [4]. Simple homological methods or advice on how to find convenient sources of them can be taken from Matsumura [31], Bruns/Herzog [6], Eisenbud [12] and Hilton/Stammbach [18]. Only few concepts of algebraic geometry are being used, especially however the concept of the projective spectrum. For basic structures of this kind we refer to Hartshorne [15]. Discourses on resultants have been abundant for a long time. A historical picture can be gathered from the papers [21], [22], [23], [24], [25] VIII Preface of Jouanolou, where many extensions are developed in terms of modern algebraic geometry. Characteristics of elimination from a more geometric standpoint and a report on the corresponding literature are provided by Gel'fand/Kapranov/ Zelevinsky [14]. The text book Cox/Little/O'Shea [9] can be recommended for a first reading. Our interest in the subject of regular sequences and resultants started in 1988 when there seemed to be no modern account of the classical treatment as, say, that given in 1913 by Adolph Hurwitz [19]. So we made up our own version of the subject starting from Hurwitz's article and generalizing to arbitrary noetherian commutative ground rings. Some notes on the subject which we have written down occasionally over the years, whenever we found time for it, are now put together for these notes. We could not deal with all of the developments of elimination theory in recent years without unduly enlarging the text. Instead we restricted our attention to those parts of the theory which are closely related to the concept of complete intersection, influenced by [41], [42], [28] and later by [43], [44]. During the preparation of these notes we were aided by many people. We are grateful to David Eisenbud and the Fellowship of the Ring at Brandeis University, Mass., for profitable discussions and general assistance during our stay there in February/March 1992. It was supported by the Deut- sche Forschungsgemeinschaft (DFG) which we gratefully acknowledge. On several occasions the second author had the opportunity to report on the progress of these notes in a series of lectures at the Indian Institute of Sci- ence at Bangalore organized by Dilip Patil and supported by the Deutscher Akademischer Austauschdienst (DAAD). Many helped us by pointing out details of interest, putting questions to us or providing us with background material. In particular, we mention Gerd Miiller, Jiirgen Herzog, Jose Gomez Torrecillas, Dilip Patil and our students Goran Devic, Benjamin Nill as well as people in our lectures and talks on the subject. We would like to express our sincere thanks to them. Last but not least, we cordially thank Wilhelm Kaup and Klaus Peters for their help in editing these notes.

Bochum and Tubingen G. Scheja and U. Storch Contents

I Preliminaries 1 Kronecker Extensions 1 2 Modules and Kronecker Extensions 6 3 Numerical Monoids 13 4 Relations of Numerical Monoids 23 5 Splitting of Numerical Monoids 36

II Regular Sequences 6 Regular Sequences and Complete Intersections 41 7 Graded Complete Intersections 53 8 Generic Regular Sequences 60 9 The Generic Structure of the Principal Component 75

III Elimination 10 Basics of Elimination 81 11 The Main Case for Generic Regular Sequences 85 12 The Main Case for Regular Sequences 96

IV Resultants 13 Resultant Ideals 105 14 Resultant Divisors and Duality 109 15 Resultants 119 16 Formulas on Resultants 129

References 137 Index 141

I Preliminaries

1 Kronecker Extensions

In this section and in the next one we define Kronecker extensions and develop their basic properties in terms of commutative algebra. Kronecker extensions provide the conceptual tools for Kronecker's method of indeterminates ("Unbestimmten-Methode"). Special cases of it have been used for a long time, cf., for instance, Supplement 7 in Section 2. A modern use of the method can be found in Nagata's book [33]. In the following A always denotes a commutative ring.1 Spec A is the prime spectrum of A and SpmA the maximal spectrum of A, both of them en- dowed with the Zariski topology. The content C(-F) of a polynomial F over A (in any set of indeterminates) is the ideal in A generated by its coefficients. F is called primitive if its content is the unit ideal. For a system Ui, i & I, of indeterminates over A the short notation will be used whenever this is not subject to misinterpretation. A polynomial F G A[U] is primitive if and only if for any maximal ideal m in A the image of F in (.A/m)[J7] is not the zero polynomial. In other words: The complement of the set S of all primitive polynomials in A[U] is the union of all the extended ideals rru4[?7], m e Spm A. 1.1 Lemma The set S of primitive polynomials in A[U] is a saturated multiplicative system of non-zero-divisors. Proof. By the last remark above, S is a saturated multiplicative system. The fact that a primitive polynomial is a non-zero-divisor simply follows from the well-known Lemma of McCoy which states: // F € A[U] is a zero-divisor then there is an element a € A, a ^ 0, with aF = 0. n We now introduce Kronecker extensions. Definition For an arbitrary family Ui, i G I, of indeterminates the A- algebra

1 A commutative ring is a commutative ring with identity. All modules are unitary.

1 2 I Preliminaries that is the ring of fractions of A[U] with respect to the multiplicative system S of primitive polynomials, will be called the Kronecker extension of A (in the indeterminates Ui, i G J). 1.2 Proposition Every Kronecker extension A —> A(UJ is faithfully flat. Proof. A —)• A(U) is the composition A —» A[U] —» A[l/]g and therefore flat. Because of mA[U] fl S = 0 for m e SpmA we have mA(U) ± A(U). Thus the extension is even faithfully flat. n Especially A(U)X D A = Ax. 1.3 Proposition The canonical mapping SpecA(U) —> SpecA induces a homeomorphism SpmA(t/) —> SpmA. Every maximal ideal of A(U) is the extension of a maximal ideal of A. Proof. SpmA(t/) can be identified with the subspace M of Spec A[U] con- sisting of those (prime) ideals which are maximal in the set of all ideals of A[U] having empty intersection with S. For <}3 e M the following is true: A polynomial F € A[U] belongs to ^3 if and only if all its coefficients belong to fp, i. e. G(F) C degF we have F + UfG G fp and A ^ G(F + UfG) = G(F) + C(G) 2aA + C(G), a contradiction. As a consequence every ?p G M is an extended ideal: where ty n A is clearly maximal in A. Conversely, rru4[J7] € M for every m G SpmA. Furthermore, for any ideal 21 in -4[J7], where a is the ideal generated in A by the coefficients of all elements of 21. Thus the projection SpmJ4(?7) —» SpmA is not only bijective but a homeomorphism, too. D As a corollary to 1.2 and 1.3 we have that Kronecker extensions preserve the length of modules. 1.4 Concordance rule Let A —> A' be an integral homomorphism of rings. Then

Proof. A homomorphism A —»A' induces a homomorphism A(U) —>A'(U) and hence a canonical homomorphism A'®AA(U) —>A'(U), which is simply 1 Kronecker Extensions 3 the inclusion where T is the canonical image of S in A'[U] and S' is the set of all primitive polynomials of .A'[J7]. We have to prove that S' is the saturated hull of T (assuming A —> A' being integral). Let $}' be any prime ideal in A'[U] with !'[[/] such that *$' C 0' and 0' n A[t/] = tru4[l/]. Furthermore, 0' is a minimal prime of rru4'[J7]. Again using the assumption that A -^ A' is integral we see that the prime ideal m' := 0' n A' D mA' is maximal in A'. Because 0' is minimal over m'A'[l/] D rru4/[J7] we have 0' = m'A'[U] and thus 0' n S" = 0, from which

Proof. This result on the codimension (= height) of prime ideals is equivalent to the corresponding statement for polynomial algebras which is easily reduced to the familiar case of a finite number of indeterminates. n From 1.5 and 1.3 we see that for a noetherian ring A every prime ideal in A(U) has finite codimension. Moreover: 1.6 Proposition A is noetherian if and only ifA(U) is noetherian. If this is the case, dim A = dim.A(J7). Proof. Assume that A is noetherian. Then A(U) clearly is noetherian if the family of indeterminates is finite. Now consider the general case and let £p be any prime ideal of A(U). ty is contained in a maximal ideal,

1 We emphasize that $p' Pi ^4 [17] denotes the pre-image of $p' with respect to the (not necessarily injective) homomorphism A[U] —> A'[U]. Similar conventions will be used throughout the notes. 4 I Preliminaries which is extended by 1.3 and has finite codimension by 1.5. Thus ?p has finite codimension, too. For every finite subset J C I we denote by ^3j the (finitely generated) prime ideal (

dimensions follows by 1.5 and 1.3. D For instance, 1.6 implies that A is artinian if and only ifA(U) is artinian. For any A-module V let

The main properties of modules related to Kronecker extensions will be studied in the next section.

Supplements 1. Let V be an yl-module. The content C(G) of a polynomial G € V[U] = A[U] ®A V with coefficients in V is defined to be the submodule of V generated by the coefficients of G. For a primitive polynomial F € A[U] the following holds: C(FG) = C(G). (The problem is easily reduced to the case of one variable by substituting the Ui by suitable powers of one variable. Moreover one may assume that A is a field.) Applications: (1) Primitive polynomials are non-zero-divisors on V[l/]. This means that the canonical homomorphism V[l/] —> V{U) is an injection. (2) Let Fj € -A[[7], j € J, be polynomials which are linearly independent modulo m for every m € SpmA If "Sj^jFjVj = 0 in V(U) for elements Vj € V then Vj = 0 for all j € J. More generally: If W is a submodule of V and Ej^jFjVj € W(U) then Vj € W for all j € J. 2. Let Vj , j € J, be an arbitrary family of submodules of an yl-module V. Then

(For an arbitrary family Wj , j € J, of A-modules the canonical homomorphism is iiijective.) Applications: (1) If o C A is a radical ideal, so is aA(U) C A(U). In particular, if A is reduced, so is A(U). (2) The extension m.AA{U) of the Jacobson radical m^ of A is the Jacobson radical m^u) of A(U). 3. Let V be an yl-module. Then

4. Let Ui, i € J, and Vj , j € J, be disjoint families of indeterminates over A. Then there is a canonical isomorphism 1 Kronecker Extensions 5

(It's easy to write down canonical homomorphisms inverse to each other.) 5. A is normal (that is: an integral domain being integrally closed in its fields of fractions) if and only if A(U) is normal. More generally: Let A be an integral domain and A' the integral closure of A in its field of fractions K. Then A'(U) is the integral closure of A(U) in its field of fractions (= K(U)). 6. Assume 1^0. For elements a, b € A the ideals aA and bA coincide if and only if a and b are associated in A(U). (If a = rb and 6 = sa, then (U+r)b = (sU+l)a.) 7. Let A be an integral domain and a, b £ A. When considering quadratic forms and their discriminants the following is useful: a and 6 differ (multiplicatively) by the square of a unit in A if and only if the same is true in A{U). (Assume a ^ 0, 6 7^ 0. Let F and G be primitive polynomials in A[U] such that aG2 = bF2. Then aA = bA, i. e. there exists a unit e in A with a = eb. We have to prove that e actually is a square in A. For that it suffices to show that F and G generate the same ideal in yl[l/]. This can be checked in yl(1/)[(7] where V-i , i € I, is a new set of indeterminates corresponding to the Ui , i € I. But from eC2 = F'2 and eG(V'f = F(V'f we get (eG(V)G - F(V)F)(eG(V)G + F(V)F) = 0.) 8. t Let A —> A' be a homomorphism of commutative rings, Ui , i € /, a non- empty family of indeterminates over A and h : A' ®A A(U) —> A'(U) the canonical homomorphism. This supplement provides some additional remarks on the concordance rule 1.4. a) Let A' := A[Xj]j£J , J ^ 0. Then h is not bijective. b) Let A —> A' be a field extension. Then h is bijective if and only if A —> A' is algebraic. (For instance, consider a field K between A and A' such that K is purely transcendental over A and ^4 is algebraic over K.) c) Let A be a noetherian integral domain and A' = K its field of quotients. Then h is bijective if and only if dirndl < 1. (Let dirndl = 1. One has to show that K ®A A(U) = K[U]s (C K (U)) is a field. Assume that there is a prime ideal 971 ^ 0 in K[U]S • Then «p := SOT n ^l[l/]s is a prime ideal 7^ 0 in A[U]S = A(U) and therefore by 1.6 and 1.3 of the form mA(U) with m € Spin A Contradiction. Let dirndl > 2. Instead of A we consider the integral closure of A in K. Then we may assume that A is a Krull domain of dimension > 2, which contains a regular sequence a, b with Aa + Ab ^ A. Then the polynomial F := a + bUi0 (for an arbitrary ig € I) is prime in A[U] but not primitive. F is not invertible in K[U\S.) d) Let A be noetherian and dirndl < 1. If for every prime ideal p' € Spec^4' the extension Q(yl/yl np') C Q(yl'/f>') of quotient fields is algebraic, then ft is bijective. In particular, h is bijective if A is noetherian of dimension < 1 and if A = AN where N is any multiplicative system in A. (Consider a prime ideal «p' C A'[U] disjoint to S. One has to show that *$A'(U) ^ A'(U). Let p' := A' n $p' and p := A n p'. Switching to residues modulo -p and modulo p' one reduces to the case p = 0 and p' = 0, i. e. A C A' is an extension of integral domains. Switching to the quotient fields K and K' = K ®A A' of A and A' one reduces to a case covered by c).) (The fact that Kronecker extensions are not universally compatible with base ring extensions means that they cannot be applied directly in a geometric context.) 2 Modules and Kronecker Extensions

As in the previous section A always denotes a commutative ring. We denote by Ui, i & I, a given system of indeterminates, by A(U) the corresponding Kronecker extension and by V(U) the extended module A(U) ®A V of an A-module V.

2.1 Theorem Let V be a locally cyclic A-module and let vi,...,vm be elements ofV. Furthermore, let Fi,...,Fm be polynomials in A[U] the residue classes of which in (A/m)[U] are linearly independent over A/m for every m G SpmA. The following assertions are equivalent: (1) t'i,..., vm generate V over A. (2) FlVl + ••• + Fmvm generates V(U) over A(U). In particular, V(U) is a cyclic A(U)-module ifV is finite over A. Proof. (I) follows from (2) using the fact that A ->• A(U) is faithfully flat. (1) implies (2). It is enough to prove that x := FiVi + • • • + Fmvm generates V(U) modulo mV(U) for any maximal ideal m of A. Thus we may substitute A by any of its residue fields k. By assumption on V, V(U) is then of dimension < 1 over k(U) and hence generated by one of the elements t'i,..., vm , say by v\. There are elements 02,..., am, of k such that Vi = Ojt'i, i = 2,..., m. Then x = Fvi, F := FI+ a^F^ -\ hamFm . By assumption on FI, ...,Fm we have F =^ 0 and hence V(U) = k(U)x. n

The assumption on FI, ..., Fm in 2.1 is equivalent to the following one: FI, ..., Fm generate a free direct A-summand of A[U] of rank m. For applications one often chooses simply indeterminates U\,..., Um for JP1,...,JPm.Then will be a generating element of V(U), a so-called general generator of V(U) in the line of Kronecker's method of indeterminates. Another way could be to use the element

with just one indeterminate U and pairwise different exponents n\,..., nm . 2.2 Corollary Assume 1^0. Equivalent are: (1) V is a finite protective A-module of rank 1. (2) V(U) is a free A(U)-module of rank 1. Proof. (2) implies (1) for any faithfully flat change of rings. (1) implies (2) simply by Theorem 2.1. n

6 2 Modules and Kronecker Extensions 7

It is worth noticing that the technique in Theorem 2.1 can be generalized in natural ways. To formulate one result in that direction we introduce the following notation: Let V be a finite A-module; then

2.3 Theorem Let V be an A-module generated by the elementsVi,... ,vm. Furthermore, let (Fij) be, a (m, x LI)-matrix with entries from, A[U]. Assume that for every s with X < s < /i there are s columns in (Fij) all the s-minors of which are linearly independent modulo m for every m G Spm A Then the p, elements

generate the A(U)-module V(U). Proof. As in the proof of Theorem 2.1 one is easily led to the case that A is a field. Then the rest of the proof is an exercise in linear algebra. n The easiest way to construct a matrix (F^) for 2.3 is to have different indeterminates Uij for Fij. On the other hand, one can always construct matrices of the type wanted by choosing for Fij suitable monomials Unij in just one indeterminate U over A. 2.4 Corollary Assume I ^ 0. Equivalent are: (1) V is a finite projective A-module of rank r. (2) V(U) is a free A(U)-module of rank r. Theorem 2.3 can be partially generalized in the following way. 2.5 Theorem Assume I ^ 0. For any finitely representable A(U)-module W the number HA(U)(W) ^s ^e wwmrooZ number of generators ofW. Proof. By induction on fj, = HA(u)(W) it is sufficient to find an element w € W such that nA(u)(W/A(U)w) = p-1 (if p > 0). Let W = A(U)wi + • • -+A(U)wm . We are going to construct a linear combination w = -F\u'i + 7 \-Fmwm , Fi € A[U], such that for every m G Spm A with dimVF/mW = /j, the residue class w G W/mW7 is ^ 0. There is, as we shall see, an integer s with the following property: For any m G Spm A with dimW/mW7 = [i, there is a J C {!,..., m} with | J\ = p, and with polynomials Gij, G G (A/m)[t/j, degGjj < s, G ^ 0 and in W/mW7. Then let F G A[U] be a homogeneous primitive polynomial of degree > s and set Fj := F^~l, j = 1,..., m. In W/mW7 the coefficient 8 I Preliminaries of w = FI-WI H h Fmwm at un is G~^f=lFjGij ^ 0 where i € J is chosen in such a way that (Gji,..., Gim) 7^ 0 (which is possible because (G«)^0)._ It remains to show how to find s, J, GJJ , G. One way to do it is to represent W7 as a residue class module of a free module A(U)m with basis ei,... ,em modulo a submodule generated by (finitely many) elements y^ = Y^=lHjkej, k = l,...,r, Hjk e A[U]. Consider any m e SpmA with dimW/mW7 = p,. There is a subset J of {1,..., m} such that Wi, i & J, is a basis of W/mW7 arid a subset K of {1,..., r} such that &;, i G J, arid j/fc , A; G .K", together form a basis of (A/m)(f/)m . Then choose for G resp. GJJ the residue classes of the determinant resp. of suitable minors of the matrix built with Cj, i G J, and j/fe, fc G -K", as columns. There are only finitely many choices. D 2.6 Corollary Assume I ^ 0. Any finite protective module over A(U) with rank is free. In particular, (finite) stably free modules over A(U) are free. The group PicA(U) of classes of finite projective A(U)-modules of rank one is trivial. Finally we broach the problem of determining algebra generators of finite commutative algebras when Kronecker extensions are applied. Consider a finite commutative algebra B over a field k. Let K be any field extension of k with infinitely many elements. Then the minimal number of Jf-algebra generators oiB(jf) = K®^B is independent of the choice of K; we denote it by and call it the true number of fc-algebra generators of I?. (That v is independent of the choice of the infinite field K follows from the fact that the number of algebra generators does not decrease when field extensions of K are applied.) For an arbitrary finite commutative A-algebra B we define

2.7 Theorem Lei B = A[xi,...,xm] be a finite commutative A-algebra and let (Fij) be a (m x v)-matrix with entries from A[U], v = VA(B). Assume that for every s = 1,..., i/ there are s columns in (Fij) all the s- minors of which are linearly independent modulo m for every m € SpmA. Then the v elements

generate the A(U)-algebra B(U). 2 Modules and Kronecker Extensions 9

Proof. Again we may assume that A = k is a field and even that k is algebraically closed. (The concordance rule is being used.) In addition, we may switch to the local components of B, i. e. we may assume that B is local with maximal ideal 971. We replace Xi by Xi — 04 , where Oj G k is the residue class of Xi, and thus may finally assume that the Xi generate 971 as a S-module. But then yl5..., yv generate 97l(f7) over B(U) by 2.3, and B(U) = k(U)^...,yv}. a In particular the situation of 2.7 can always be realized using a Kro- necker extension by one indeterminate. 2.8 Corollary Let B be a finite commutative A-algebra. If I ^ 0 then the A(U)-algebra B(U) can be generated by VA(B) elements. For a generalization see Supplement 9.

Supplements

1. Let FI,. ..,Fr € A[U] be polynomials such that G(Fi) + • • • + G(Fr) is a projective ^4-ideal. Then

Moreover, if I ^ 0, this ideal is principal (generated by F = GiFi + • • • + GrFr where GI, ..., Gr are suitably chosen monomials; use Theorem 2.1). 2. (Prefer and Bezant domains) Let A be an integral domain. A is called a Priifer (resp. Bezout) domain if finitely generated ideals in A are projective (resp. free) ^4-modules. If I ^ 0 then the following assertions are equivalent: (1) A is a Priifer domain. (2) A(U) is a Bezout domain. (3) A(U) is a Priifer domain. Tf these equivalent conditions are fulfilled then every finitely generated ideal of A(U) is extended from A. 3. (Dedekind domains) Assume that 1^0. Then the following assertions are equivalent: (1) A is a Dedekind domain. (2) A(U) is a principal ideal domain. (3) A(U) is a Dedekind domain. If these equivalent conditions are fulfilled, o i-y aA(U) is an isomorphism from the lattice of ideals of A to the lattice of ideals of A(U). 4. (Frobenius algebras) Let B be a finite projective commutative ^4-algebra. B is called a quasi-Frobenius (resp. Frobenius) algebra over A if the relative dualizing module E := HomA(B,A) is a projective (resp. free) B-module (necessarily of rank 1). If I ^ 0 then the following assertions are equivalent: (1) B is a quasi-Frobenius algebra over A. (2) B(U) is a Frobenius algebra over A(U). (3) B(U) is a quasi-Frobenius algebra over A(U). (HoiuA(u)(B(U),A(U)) = E(U).) 10 I Preliminaries

5.t Let A be a Krull domain, 1^0. Then A(U) is a Krull domain and the canonical map 01 (A) —> 01 (A(U)) of divisor class groups is surjective with kernel Pic .A, cf. [4], Oh. VII, § 1, no. 10, i.e. there is a canonical exact sequence

In particular, PicA(U) is trivial. A(U) is factorial if and only if A is locally factorial. (PicA(U) = 1 follows also via 2.6.) 6. t The construction of Kronecker extensions can be generalized in an obvious way. As before let Ui , i € I, be a system of indeterminates over A and assume I 7^ 0. Let Y be an arbitrary subset of Spec A. Then the set S = SY of polynomials F € A[U] with C(-F) 2 P f°r every p € Y is a saturated multiplicative system in -A[C7] (containing the system of primitive polynomials). The ring of fractions -A[f7]s will be denoted by Ay(U). It is flat over A. The space Spm Ar(f7) consists of the prime ideals pAy(U) where p € Spec A is maximal with respect to the following property: Any finitely generated subideal o C p is contained in some prime ideal of Y. If A is noetherian then Spm Ay (U) consists of the prime ideals $AY(U) where p is maximal in Y. The analogon of Theorem 2.5 holds. For Y = Spec A we get the Kronecker extension A(U) of A. 7. t Let A be a Krull domain which is not a field and assume 1^0. By Yi we denote the prime ideals of codimension 1 in A. The polynomials of SYI are called 1-primitive. Then the ring ^1^(17), defined in Suppl. 6, is a principal ideal domain, and the canonical homomorphism Div (A) —> Div (-Ayi(J/)) of groups of divisors is bijective. For a Dedekind domain A we have, of course, Ay1 (U) = A(U), cf. Suppl. 3 above. (This construction goes back to Kronecker and was described more precisely by J. Konig in [27], IX, §§7,8, who thereby gave an interpretation of Kummer's ideal numbers in terms of polynomials: A divisor d € Div (A) is represented by a polynomial F € A[U] which defines the divisor d in Div (Ay1 (U)). The construction of AYI (U) should not be confounded with the construction of j4[C/]p where P is the multiplicative system generated by the prime elements of yl[l/]. ^l[l/]p has the same divisor class group as A and is a Dedekind domain (or a field) if I 7^ 0. This depends mainly on the following statement the proof of which we indicate for the sake of completeness: Every prime ideal ?p C A[U]jI 7^ 0, of codimension > 1 contains a prime element. We may assume right away I = {!}. Let p := %$r\A. Then codimp > 1. If codimp > 1 and a, 6 € p are chosen coprime n a + bU € $p is prime. Now let codimp = 1. Choose F = ao+aiU + • • • + anU € ty such that F generates ${L4p[£7]/p,Ap[[/], an <£ P- Let pi,...,pr be the prime ideals of codimension 1 in A containing ao, ai,..., an and pi,..., pr,..., ps those containing ai,..., an • If a € pr+iPl- • -npsnp , a ^ piU- • -Upr , then G = a+F € ?p is prime because G is 1-primitive and prime over the quotient field K of A. (Any decomposition of G over K induces a decomposition in ^4p[l/] modulo p^4p[l/].)) 8. t Let A be a Krull domain, K its field of fractions, L a finite extension field of K and B the integral closure of A in L (a Krull domain, too). A polynomial F € B[U] is 1-primitive if and only if its norm NALJ(.F) is 1-primitive. (For the definition of 1-primitive polynomials see Suppl. 7. The proof is done by localizing A with respect to prime ideals p of codimension 1 in A and looking at the Kronecker extensions Ap(U) C Bp(U). 2 Modules and Kronecker Extensions 11

The criterion given here served Kronecker and Konig to define 1-primitive polynomials over B in case that the subdomain A is factorial, thus reducing the concept to the one of Gaufi for polynomials over factorial domains.) 9. Assume 1^0 and let C be a finite and finitely represented commutative ,A([/)-algebra. Then VA(U) (C) is the minimal number of ^4(l/)-algebra generators of C. (The proof runs along the same lines as that of 2.5.) 10. t Let B be an arbitrary finite commutative ^4-algebra. The invariant v = VA(B), which we referred to in Theorem 2.7, can be computed using the module OA(S) of Kahler differentials: One has v = rasx(l,/J,B(^A(B))) if the structure homomorphism A —> B is not surjective, and v = 0 otherwise. (For the proof we may by definition of v assume that A = k is an infinite field. Then the result is Lemma (4.2) in [37]. For the reader's convenience we give an outline of the proof. First we may assume that k is even algebraically closed. Looking at the local components of B and using the Chinese Remainder Theorem we may furthermore assume that B is a local fe-algebra. But in this case

11. Assume 1^0. Let B = A[XI, ..., xm] be a finite commutative ^4-algebra and n a positive integer. For every m € Spinal assume that the fibre B/mB has local components of embedding dimension < n and residue fields which are separable over A/m. Then we have VA(B) < n, and there are linear combinations j/i,..., yn of Xi,..., xm with coefficients in A[U] such that B(U) = A(U)[yi,..., yn}. 12. t Let B be a finite projective commutative ^4-algebra. B is called separable (or etale) over A if B is unramified over A, i.e. if the module QA(B) of Kahler differentials is the zero-module (or equivalently, if E = Hona.A(B,A) is generated as a B-module by the ordinary trace tr ^, or again equivalently if the fibres A/m. —> B/mB are separable for every m € Spmyl). To have a smooth formulation in the following let B have rank T over A. If B is separable over A, so is B(U) over A(U). If furthermore 1^0 then by Suppl. 11 the j4(C/)-algebra B(U) has a primitive element, i.e. there is a monic (separable) polynomial a. of degree r in one indeterminate X over A(U) such that B(U) = A(U)[X]/(a).

13. t Let B = A\XI,..., xm] be a finite projective commutative ^4-algebra with rank and FI, ..., Fm polynomials in -A[C7] which are linearly independent modulo m for every m € Spinal. (For example, Fi := Ui , or Fi := Un' with just one indeterminate U and pairwise different exponents rii,..., nm .) Let and denote the characteristic polynomial of x by x € A(V')\_X\. For every m € Spm A such that the residue fields of B/mB are separable over A/m let be the canonical prime factor decomposition of the residue class x °i X in the polynomial ring (A/m)(U)[X}. Then the monic prime polynomials Pk are the minimal polynomials of x in the residue fields of (B/mB)(U) over (A/m)(U) and the 6fe are the multiplicities of the corresponding local components of (B/mB)(U), which, of course, have the same multiplicities as the local components of B/mB , in which one might have been primarily interested. (One can assume A = A/m 12 I Preliminaries and B = B/mB local. In this case x = Pe'> where P is the minimal polynomial of the residue class of x in the residue field of B, which generates this field by Theorem 2.7. In general, the separability condition is necessary: Let if be a field of prime characteristic p such that there is an element a € K^KP. Consider B := K[X, Y]/(XP, Yp - a) over A := K.) 3 Numerical Monoids

The discussion of the grading of a polynomial algebra .A[To,..., Tn] given by positive weights 7oi---i7n of the indeterminates TQ, ...,Tn natu- rally leads to questions about the monoid generated by the weights. There- fore, in this section and the next one we will introduce some concepts and propositions about (additive) numerical monoids. For general as well as for historical references we refer to [5], [13] and [2]. We assume throughout that the given positive weights 70, • • •, 7« are relatively prime, i. e. gcd(7o,..., 7n) = 1. Then they generate a numerical submonoid of N which shall be denoted by

Every numerical submonoid of N can be described in such a way. Note that a submonoid M of N is called numerical if and only if the set N\Af of gaps is finite. Then is called the degree of singularity (or the genus) of M. (Usually this number is denoted by SM which we are to avoid, however, because we reserve $ for the degrees of polynomials.) A numerical submonoid M of N defines the monoid algebra .K"[Af] over an (arbitrary) field K, which is the KXm of the graded subalgebra £m€M polynomial algebra K[N] = K[X] with degX = 1. The ring .K"[.X] is the normalization of .K"[Af], and "&M = &imi(K[X]/K[M] which motivates the expressions "degree of singularity" (and "genus") for -&M • To ring-theoretic concepts concerning the ideals of .K"[M] correspond concepts about ideals of M. Let M C N be a numerical submonoid of N. A subset I C Z is called an M-ideal if J\N is finite and if M + I C I, where M + I = {a + x : a € M,x G /}. The empty ideal will be called the zero-ideal. An ideal J 7^ 0 is also called a fractional ideal of M. The fractional ideals of M correspond bijectively to the homogeneous fractional ideals of .K"[Af] in its graded quotient ring K[X, X~1]. To the usual operations a + b, a n b, a • b and a : b of fractional ideals a, b over .K"[Af] correspond the operations lUJ, IflJ, J+J and of M -ideals J,J. The maximal ideal of M is

13 14 I Preliminaries

For any M-ideal J the finite set /\(Af+ + J) is the uniquely determined minimal set of generators of J as an M -ideal. The cardinality of the minimal set E := Af+ \ (Af+ + Af+) of generators of the maximal ideal is sometimes called the embedding dimension of M. The set E is also the uniquely determined minimal set of generators of the numerical monoid M itself. Proof: Of course, any subset of M which generates M contains E. To prove M = Mon(.E), assume the contrary and consider the minimal element c € M\Mon(.E). Then c ^ E, hence c = a+b with a, b € M+ . Being smaller than c, the elements a,b must be in Mon(.E). Hence c G Mon(.E), too, a contradiction. The ordinary dual of an ideal J 7^ 0 is [M — I]. As the adj oint ideal (or simply adjoint) of an M -ideal J 7^ 0 we define the M -ideal

Obviously, J" = J. The ideal is called the dualizing ideal (or canonical ideal) ofM. (The corresponding ideal over .K"[Af] is the dualizing module (or canonical module) ^K[M} of the graded algebra .K"[Af]; cf. [6], Section 3.6.) Then for all Af-ideals 1^0. For any ideal J ^ 0 let

be called the order of J. Then is the so-called Frobenius number of! which is nothing else but the maximum in the set of integers not belonging to I. For M = Mon(7) we will also write for the Frobenius number and the degree of singularity of M. By a singularity degree can be defined for I, too. In case i?/ > 0, g/ is just the maximum of (N + v/)\J. The set 3 Numerical Monoids 15 i. e. the minimal set of generators of the adjoint of the ideal J ^ 0, will be called the type set of I. (Some authors take the set —T(J) for the type set of J.) Then —g/ is the smallest element in T(J). The cardinality of T(J) is the so-called type of J, denoted by t/ . If tM = 1) i-e- if the dualizing ideal OM is a principal ideal, then M is called symmetric. (This means that Jf[M] is a Gorenstein ring. See also Supplement 1.) The characteristic function of an M-ideal / is the Hilbert function of P the corresponding homogeneous ideal XlPe/ KX '. Its generating function is called the Poincare series (or Hilbert s e r i e s) of J. This series is always a rational function of type

with a Laurent polynomial Qi G Z[Z, Z 1]. If J ^ 0, then £// is the polynomial with the lower degree v/ and the upper degree g/ + 1. Note that

Another consequence is the equation where £// is the derivative of the Laurent polynomial £// > in particular 0M = G'M(1) = G'M(1)/GM(1). Furthermore,

Another way of presenting "P/ is to choose an element m G Af+ and to compute the numbers for i = 0,..., rn — 1. Then

because of J = I+^—Q (I (~l (Zm + i)), which implies 16 I Preliminaries

These formulas are often used to compute Frobenius numbers and degrees of singularities, see Supplement 6 for an example. The family m, i = 0,..., m — 1, is called the Apery basis or standard basis of/ with respect to m. The Frobenius number and the degree of singularity of the monoid M do not depend on a special representation M = Mon(7), 7 = (70,..., 7n). The opposite is the case for the concepts we turn to, now. One should keep in mind, though, that the set f7o,...,7n} always contains the uniquely determined minimal set of generators of M. For aeZn+1 let

The sum 70+ • • • +7« = {(!,...,!),7} will be used in several formulas. For convenience we abbreviate it by

For m G Z, denotes the set of all (n + l)-tuples of non-negative integers representing m. Then

is a subgroup of the group

of all syzygies of 70,...,7n . In case [F : F(m)] < oo we call m a sating number (with respect to 7). The set is an Mon(7)-ideal since F(m) C T(x + m) for x & Mon(7), m € N. The ideal Sat(7) is not empty because F(m) = F for m := lcm(7o,..., 7n). The Frobenius number gsat(7) °f Sat(7) will be abbreviated by

In case F(m) = F, we call ma very sating number (with respect to 7). The set is also a non-empty Mon(7)-ideal. Its Frobenius number gysat(7) will be denoted by vsat7. 3 Numerical Monoids 17

Always VSat('y) C Sat (7) and consequently sat7 < vsat7. This inequality may be strict, e. g. for 7 = (6,8,9,13) one has sat7 = 31 and vsat7 = 32. Another example is provided by 7 = (6,7,9,17). For any 7 = (70,..., 7n) with n < 2, the equality sat7 = vsat7 holds, see Supplement 14. More generally, for arbitrary r G N the M-ideals can be defined. The Frobenius numbers of these ideals will be denoted by 0 r sat7. One has Sat (7) = Sat (7) and Sat (7) = M for r>n.

3.1 Proposition Let n > 1. For 7 = (70,..., 7n) and M := Mon(7) the following holds: (1) VSat(7) C Sat(7) C M + % C M+ for i = 0,... ,n. (2) g7 + maxj7i < sat7 < vsat7 < g7 + max^ 1011(7,, 7,-). Proof. To prove (1) consider m G Sat(7). There is an a = (aoi • • • i^n) in Nm(7) such that m = (0,7) and on > 0; otherwise F(m) would contain only syzygies (oo,...,an) with Oj = 0, which contradicts the condition [F : F(m)] < oo. Thus m - 7* € M. The first two inequalities in (2) follow from (1). To prove the last inequality, observe first that F is generated by the regular syzygies 7jej —%€j , n+1 i ^ j, where CQ, ..., en is the standard basis of Z . Let h denote the maximum of the hij := lcm(7j, 7,-), 0 < i < j < n, and consider any m > g7 + /i. For a special pair i < j there is an element a G Nn+1 with m — hij = (a, 7). But then and the difference of these two representations of the number m yields even gcd(7j, 7j)^1(7jej — %ej) G F(m). Therefore F(m) contains the regular syzygies. n

The following proposition will be helpful in the handling of sat7 and vsat7.

3.2 Proposition For 7 = (70,..., 7n) and 7 := (70,... ,7«,7n+i) with 7n+i G M := Mon(7) the following holds:

(1) Sat(7) = Sat(7) n (M + -jn+i) , sat7 = max(sat7 , g7 + jn+i). (2) VSat(7) = VSat(7)n(M + 7n+1) , vsat7 = max(vsat7, g7+7n+1).

In particular. Sat (7) = Sat (7) if and only if Sat (7) C M + 7n+i, and VSat(7) = VSat(7) if and only ifVSat(j) C M + -jn+i- Proof. The canonical inclusions Syz(7) C Syz(7) and F(m;7) C F(m;7) induce a homomorphism 18 I Preliminaries

We fix a representation 7ra+i = £o7o+' • '+£n7n with Si G N. Then Syz(7) = n+1 Syz(7) ® Zs with s := (EO, ... ,en, —!)• The homomorphism Z —» Z™ with (OQ, ... ,on,on+i) M- (ao + £oa«+i>- •• ,an+enon+i) maps Syz(7) into Syz(7) and Nm(7) into Nm(7), therefore F(m;7) into F(m;7), and hence induces a homomorphism

Because ofgoh = id the homomorphism h is injective. This proves Sat(7) C Sat(7) and VSat(7) C VSat(7). With 3.1(1) we get the inclusions Sat(7) C Sat(7) fl (M + 7n+i) and VSat(7) C VSat(7) n (M + 7n+i). To prove the opposite inclusions it suffices to show that for m G Af + 7n+i the homomorphism /i is surjective. But an element m = ao7o + '' •+an7n+7n+i has also the representation m = (c*o + £0)70 + • • • + (an + en)7n • Therefore a G F (77157) which means that h is surjective. The additional formulas about Frobenius numbers follow from the general formula for arbitrary fractional M-ideals /, J. In some cases monoids can be reduced to simpler ones in a natural way. The most important one of these techniques is the so-called reduction in codimension 1, originated by S.M.Johnson in [20], which we are going to describe now. Another one will be studied in Section 5. Again, let 7 = (70,..., 7«), n > 1, generate the numerical monoid M = Mon(7). Let us assume that d G N* is a common divisor of 71,..., 7n . Then gcd(7o,d) = 1. Let and M' := Mon(7'). The structures of M and M' can be compared with each other in an easy way using the canonical reduction map which we define as follows: Let m G Z. There is a unique integer s(m) such that 0 < S(TTI) < d and m = s(rn)7o modulo d. Then

The mapping R is surjective. Each of its fibres consists of d elements. Furthermore, R(m + dz) = R(m) + z for all z € Z. Because of dM' C M, R maps every fractional ideal of M onto a fractional ideal of M'. For every m G Z the canonical mapping 3 Numerical Monoids 19 is bijective. Proof. Because of OQ = qd + s(m) by ordinary Euclidian divi- sion, the mapping is well-defined and obviously injective. It is surjective: Let (a'0,..., a'n) € Nfi(m)(y) be given. Then m = s(m)~f0 + dR(m) = s(m)7o + da()7o + • • • + da'n^'n , such that (ad,..., a'n) is the image of (o6d + s(m),ai,...,a4) € Nm(-y). As a consequence, m G M if and only if R(m) G M'; in particular M' = R(M). Another consequence is, that the canonical isomorphism induces an isomorphism F(m;7) —>r(J?(m);7'), for it maps the set a—/3, a, /3 G Nm(7), of generators of F(m; 7) to the corresponding set of generators of F(J?(m);7'). It follows that there is a canonical isomorphism and that J?(VSat(7)) = VSat(7'), J?(Satr(7)) = Satr(7') for all r G N. A fractional ideal I of M will be called J?-complete, if I = R^l(R(I)}. Explicitly this means: x = s(x)^fo + dx' belongs to J if and only if all the elements ^70 + dx', 0 < i < d, belong to J. This implies

for the Poincare series of J and R( J) and their numerators. By straightforward calculations one has the following applications: 3.3 Proposition Let I be an R-complete fractional ideal of M = Mon(7) and R(I) the corresponding ideal of M' = Mon(7'). Then

In particular.

The special cases in 3.3 follow from the fact that the ideals M, Satr(7) and VSat(7) are JJ-complete, a consequence of r/F(m; 7) = F'/r(J?(m); 7') for all m G Z. 20 I Preliminaries

Supplements

1. Let M be a numerical monoid. a) The following conditions are equivalent: (1) M is symmetric, i. e. IM = 1 • (2) For every x € Z, x ^ M, there is an element w € M with x + w = gM • (3) For every x € Z either x or gM — x belongs to M. (4) The following mapping is bijective:

(This motivates the notation "symmetric".) b) gM 5: 2i?M — 1 • Equality holds if and only if M is symmetric. 2. (J.J. Sylvester) Let 70,7i € N* be relatively prime and let M := Mon(7o,7i). Then gM = 7o7i —70—71 and M is symmetric. (For every x € Z there are integers a, 6 such that x = 070 + 671 with 0 < a < 71 — 1.) 3. Let 7 = (70,71,72) with gcd(7o,7i,72) = 1 and let K := 1011(70,71,72), d '.= gcd(7o,7i). For every m € N* the following holds: dm or K — dm, belongs to Mon(7o,7i). (Reduce to the case d = 1.) 4. Let M be a numerical monoid and I a fractional ideal of M. a) (&r + vj-) + ($/ + v/) = 1. (Duality theorem) (Multiplication by — 1 maps {z : z > v/-, 2 ^ I"} = {z : z > —g/ , —2 € 1} onto {M : « e I, vj < « < g/}.) b) 'i?nM < I?M • Equality holds if and only if M is symmetric. (Follows from the inclusion —gM + M C OM • The inequality $/ < I?M holds for every fractional ideal I.) 5. Let M be a numerical monoid. The following conditions are equivalent: (1) M is symmetric. SM (2) 0M = -gM + M. (2') PaM = Z~ PM • (3) PM = ~Z^''PM(1/Z). SM+1 (4) QM = Z QM(1/Z), i.e. QM is a self-reciprocal polynomial. 6. (Arithmetic progressions. A.Brauer, E.S.Selmer et al.) Let a,b,n be positive natural numbers with gcd(a, 6) = 1. The sequence 7i := a + ib, i = 0,..., n, generates a numerical monoid M := Mon(7o,..., 7n). Let a — 2 = qn + r, q € Z, 0 < r < n. Then gM = (a-l)(b + q) + q, 2i?M - 1 = gM+r(g + l). ( 7s+J7n = (l+j)a+(*+Jn)& f°r l5:*5:n>0

g7 < lcm(7o, 71) H h lcm(7o, 7n) - 7! • 3 Numerical Monoids 21

(For TO € Z there is a representation TO = ao7o — fli7i — • • • — an^/n over Z with 0 < <2i7i < lcm(7o, 7i) for z = 1,..., n.) 9. Let 7 = (70,...,7n) with gcd(7o,...,7n) = 1 and 70 < 71 < • • • < 7n • Then

g7 < 2$7 — 1 < 7o7n — (70 + 7n), sat7 < vsat7 < 27o7n — (70 + 7n) • (The inequalities not yet known are proved by induction on n, reducing first to the case that gcd(7o,... ,7n-i) = 1.) 10. Let 7 = (70,... ,7n) with gcd(7o,... ,7n) = 1, 7 := (70, • • • ,7n,7n+i) with any 7n+i € N* and M := Mon(7). Then

Furthermore, Sat(7) 3 Sat(7)n(M+7n+i) and VSat(7) 3 VSat(7)n(M+7n+i), such that sat7 < max(sat7 , g7 +7n+i), vsat7 < max(vsat7 , g7 +jn+i) • (The homomorphism h from the proof of 3.2 is for every TO € M n (M + 7n+i) surjective.) 11. Let the numerical monoid M be related to a numerical monoid M' by the process of reduction in codimension 1. Then M is symmetric if and only if M' is symmetric. 12. a) Let 7 = (70,71) be a pair of relatively prime positive integers. Then S-j = 7o7i — (70 + 7i), sat7 = vsat7 = g7 +7071 = 27071 — (70 + 7i). (See also Suppl. 2.) b) More generally, for arbitrary n > 1, let OQ,. .. ,an be pairwise relatively prime positive integers, 6 := CI,Q • • • an their product and 7 := (70,..., 7n) with 7$ := b/ai. Then g7 = n6 - |7|, sat7 = vsat7 = g7 + 6 and g7 = 2t?7 — 1, i.e. Mon(7) is symmetric. (Reduction in codimension 1. Remark: Mon(7) is even a complete intersection, cf. Suppl. 7 in Sect. 4.) 13. t Let 7 = (70,..., 7n), n > 1, generate the numerical monoid M = Mon(7), let d € N* be a common divisor of 71,... ,7n , finally 70 := 70 , 7i •= 7i/d for i = l,...,n, 7' := (TO i • • • i Tn) and M' := Mon(7'). Beside the reduction map R there is another (canonical) map R~ : Z —y Z which is defined as follows: Let TO € Z. There is a unique integer S(TO) such that 0 < S(TO) < d and TO = — s(m,)7o modulo d. Then

FT is surjective. R~(m + dz) = R~(m) + z for all z € Z. Thus 1?" maps every fractional ideal of M onto a fractional ideal of M'. a) JT(-z) = --R(z) for all z € Z. b) A fractional ideal I of M is called IT-complete if I = (R~T1(R~(I)). Explicitly this means: x = — s(x)7o + dx' belongs to I if and only if all the elements —«7o + dx', 0 1, M is not -R~-complete. The ideal OM is always /T-complete. In addition: I is -R-complete if and only if I~ is /T-complete. In this case R~(I~) = R(T)~. c) Let I be a fractional ideal of M. If I is -R-complete, then R maps the minimal set of generators of I bijectively onto the minimal set of generators of R(I). An analogous result holds for R' : I —y /T(I), if I is /T-complete. There are natural applications to the type sets and the types of monoids. 22 I Preliminaries d) Let I be a fractional M-ideal. If I is -R-complete,

If I is /T-complete, e) For every /T-complete fractional ideal I of M the following holds:

14. t Let 7 = (70,... ,7n) generate the numerical monoid M = Mon(7). a) If n > 1, then, for every m € Satn^1(7), the group r(m;7) contains a direct summand of rank 1 of Syz(7). (Let m € Sat"^1(7). There are two different representations m = (0,7) = {/3,7). If t := gcd(/3 - a) then /3' := a + \((3 - a) = (1 - j)a + \(3 is an element of Nm(7) and /3' — a belongs to a basis of Syz(7).) b) If n > 2, then, for every m € Satn~2(7), the group F(m;7) contains a direct summand of rank 2 of Syz(7). (One uses the following Lemma: Let a(0),a(l),a(2) € Nm(7) be such that x := Q(l) — a(0), j/ := a(2) — a(0) are linearly independent. Then x, j/ is part of a Z-basis of Syz(7) if and only if the closed triangle S with vertices a(0), a(l), a(2) in Qn+1 contains no other points of Zn+1 but a(0), a(l), a(2). Proof. Using the reflection z i-y —2+a(l)+a(2) the condition on S is seen to be equivalent with the condition that the parallelogram with vertices a(0), a(l), a(2), a(l) + a(2) — a(0) contains no other points of Zn+1 but its vertices. This is equivalent to the fact that x, y form a basis of the two-dimensional lattice (Q x + Q y) n Zn+1 . Let m € Satn^2(7). Choose a(0), a(l), a(2) € Nm(7) in such a way that a(l) — a(0), a(2) — a(0) are linearly independent and that the triangle S with vertices a(0), a(l), a(2) has minimal volume. Then S contains no other points of n+1 n+1 n+1 Nm(7) = N n {z e Q : (z, 7} = m} and therefore no other points of Z but Q(O), a(l), a(2). Now apply the Lemma.) In particular, if M can be generated by < 3 elements, then Sat(7) = VSat(7) and sat7 = vsat7 . 4 Relations of Numerical Monoids

Let M = Mon(7) be a numerical monoid (C N) with a given system 7 = (Toi • • • >7n) of positive generators. For every commutative ring A the canonical surjective monoid homomorphism with a i-» (a, 7} gives rise to a homogeneous A-algebra homomorphism which maps the indeterminate Tj of degree % to 7^ = Xr', i = 0,... ,n. The image of it\ is ATM]. Its kernel is called the ideal of relations of M with respect to the representation M = Mon(7). A relation of M itself (with respect to 7) isapair (a,(3) € Nn+1xNn+1 with (a, 7} = ((3,7). To such a relation of M there corresponds the binomial Ta — T@ in the ideal of relations X The binomials of this type generate the ideal 3. Proof. Because 3 is homogeneous let us consider a homogeneous a T (k) polynomial F = J2k k " G 3 of degree m > 0, ak € A, a(k) G Nm(7). m From 0 = ^A(F) = Q\ ofe)X we get £fe ak = 0. Therefore, for a fixed a T k) (K) index K, F = Y.k k( ' - T" ). The following remark is a simple application of the last result (in case A ^ 0): The M-ideal {m € M : 3m ^ 0} coincides with the M-ideal Sat™~1(7) introduced in Section 3, and the degrees of any set of homogeneous generators of 3 generate Sat™^1(7). The set of all relations of M is a compatible equivalence relation on Nn+1. In general, an equivalence relation R on the monoid Nn+1 is called compatible if (a,/3), (a',/3') € -R implies (a + a',/3 + /3') £ R. For a compatible equivalence relation R the quotient Nn+1/J? carries a unique monoid structure such that the canonical projection Nn+1 —)• Nn+1/J? is a monoid homomorphism. Let (a(j), (3(j))j j G J, be any system of relations of Af. This system generates a smallest compatible equivalence relation R containing all the (a(j), (3(j)) and 7T7 induces a surjective monoid homomorphism Nra+1/J? —> M, which is an isomorphism if and only if R is the set of all relations of M. In this case (a(j),(3(j)), j G J, is called a system of generators of all relations of M.

23 24 I Preliminaries

4.1 Proposition Let (a(j),(3(j)), j G J, be a system of relations of M and let R be the smallest compatible equivalence relation onNn+1 containing all the (a(j), /3(j)). Then the kernel 21 of the surjective algebra homomorphism

induced by the canonical projection Nn+1 —)• Nn+1/J? is generated by the binomials Ta^ - T13^, j € J. Proof. Let 21' C 21 be the ideal generated by the binomials T"(j) - T^ and for a € Nn+1 let ta denote the residue class of T" in A[T]/W. Then the monoid homomorphism a >—)• i", a G Nra+1, induces a monoid homomorphism on Nn+1/JJ which defines an algebra homomorphism A[Nn+1/JJ] —» A[T]/2l'. The composition with A[T] -> A[Nn+1/J?] is the canonical projection A[T] ->• A[T]/2l' which implies that 21 is mapped to zero in A[T]/2l', i. e. 21 C 21'. a 4.2 Corollary Let A ^ 0. A system (a(j),(3(j)), j € J, of relations of M generates all the relations of M if and only if the corresponding homogeneous binomials Ta^ —T^\ j G J, generate the ideal 3 = kerTr^ C A[T]. Note that the criterion in 4.2 is independent of the coefficient ring A =£ 0. Because for a noetherian ring A the ideal 3 is finitely generated, this is true for an arbitrary A. Therefore, any minimal set of generators of 3 is finite. The minimal number p(3) = HA[T}(^) of generators of 3 is independent of the ring A(=£ 0). Every minimal set of binomials generating 3 contains exactly p(3) elements. Proof. Let Fj , j G J, be a minimal system of generators of 3 C .A[T]. Then this system generates the kernel 21 of TT^- for any residue class field K of A. card J is greater than or equal to the dimension dinix3/9?0, gjj ;= .K"[T]+ . By the Lemma of Krull-Nakayama for graded Ji"-algebras, any minimal system of homogeneous generators of 3 contains exactly dini^3/9713 elements. Now assume that the Fj are homogeneous binomials. There is a subset J' C J such that Fj , j G J', generate 5 minimally. But J' = J by 4.2. Thus card J = dimK3/97D = p(3) in this case. Let A = K be a field. Then where

: dm (7) = dinix(3/97D)m , is the Poincare series of 3/970, which is actually a polynomial. (As the proof above shows, also the polynomial 7£7 is independent of the field K. Even for an arbitrary ring A ^ 0, dm(^f) is the rank 4 Relations of Numerical Monoids 25 of the free A-module (a/9?O)m, 3 = ker-K\ , 971 = A[T}+ . For a purely combinatorial description of Tt^ see the remarks following Proposition 4.3 below.) A system (a(j),(3(j)), j G J, of relations of M = Mon(7) will be called a minimal system of relations, if the binomials T"--1' — T^--1', j G J, form a minimal system of generators of X The number is independent of the representation M = Mon(7) of M because for any field K the number p(3) — n = /i(3) — (n +1) +1 is the deviation ("Abwei- chung") of the one-dimensional local ring K\M\ = if [TJ/MTJT]] which is the completion of Jf[M] with respect to its homogeneous maximal ideal. (For the deviation of a noetherian local ring cf. Section 6, Supplement 5.) In the situation here an elementary proof is possible, see Supplement 4a). We will call devM the deviation of the numerical monoid M. For simple reasons devM •> 0, see Supplement 1 or use the following argument: If (u(j),/3(j)), j G J, is a system of generators of all the relations of M, then the differences a(j) — /3(j), j G J, generate the group Syz(7) of syzygies of 7, which is a group of rank n. Thus | J| > n. In case devM = 0, the monoid M is called a complete intersection. We mention that M is a complete intersection if and only if for every noetherian ring A =£ 0 the monoid algebra A[M] is a graded complete intersection over A. For details see Supplement 2. As our next topic we propose to study where the minimal binomial generators of the ideal 3 = ker T{\ are situated. For this let again A = K be a field. Then we have to determine bases of the Jf-vector spaces (3/9?O)m = 3rn/(m3)rn, m G N, m := K[T}+ . It is more convenient to study the spaces K[T]m/(yjd)m and look out for their monomial bases. a fjl (3CH2r)m is generated over K by the binomials T ~T with a, j3 € Nm(7) and gcd(T",T^) ^ 1, i.e. supp(a) n supp(/3) ^ 0. Here the support of any a G Nn+1 is defined to be the set

This description of the generators of (9?l3)m suggests to introduce the following graph structure on the set Nm (••/): Two points a, /3 G Nm(7), a ^ /9, are joined by an edge if supp(a)risupp(/3) ^ 0. Let the set of the connected components be denoted by and its cardinal number by 26 I Preliminaries

A polynomial E^PN c ) ^aT" G K[T]rn belongs to (970)m j/ and only if a Eaes « = 0 for every connected component S G Nm (••/). This is a well- known and simple result of graph theory. The proof runs as follows: The a binomial generators T — T@ of (97l2f)m fulfil these relations, so do all elements of (97l3)m . For the converse it is sufficient to prove that any a a a polynomial F = J2aes aT with Y^aes a = 0 belongs to (9JO)m . We fix an element /3 G S. Then F = Eaesa«(Ta ~ T/3)- Tt suffices to show that a 13 T - T € (Ona)m . There is a chain /3 = a(0), a(l),..., a(r) = a such that a(i) G Nm(7) and suppa(i) n suppa(i + 1) ^ 0 for i = 0,..., r — 1. 3 Ta(i+1) T (i) e Then T« - T/ = EI=o( ~ " ) (W3)m • A monomial basis of K[T]m/(9Jl3)m is given by (the residue classes of) a T W), where a(j) € Nm(7), j = 1,..., cm(7), is a full system of representatives for Nm(7). Therefore we have:

4.3 Proposition Letm € N. Then3m ^ (3Jl?)m ?/ond on% ifcrn(~{) > I. In this case a basis of3m modulo (97l2f)m can be given by the binomials

where a(l),... ,a(cm(7)) is a full system of representatives for the set Nm(7) of the connected components o/Nm(7).

A similar description of 3m/(StfO)m is given in [7]. The connected components of Nm(7), m > 0, can be described in a simpler way. To do this, define

Then the sets IJQ,esSupp(a), S G Nm(7), obviously form a partition of Gm (7) into non-empty subsets of Gm (7). Let

denote the set of elements of this partition. Then cardGm(7) = cm(7), and the elements of Gm (7) are the connected components of the following graph on the set Gm(7) as set of vertices: Two points r, s G Gm(7), r ^ s, are joined by an edge if and only if there is an a G Nm(7) with r, s G supp(a). The simple proof is left to the reader. The set of m G N such that Gm(7) = {0,... ,n} and cm(7) = 1 is a subset of M and even a non-empty ideal of M, which we shall denote by

Its Frobenius number we denote by con^ . Then 4 Relations of Numerical Monoids 27

because for m > g7 + 70 + • • • + 7« there is an a G Nm(7) with supp(a) = {0, ...,n}. Note that every m G Af, which is the degree of an element of a minimal system of homogeneous generators of 3, does not belong to Con(7); in particular, rmax^ < con7 , where denotes the supremum of the degrees of the elements of such a system. See Supplements 5, 6 below and Supplement 1 in Section 5 for further details. m The coefficients of the Poincare series 7£7 = Xlmew dm(l)Z of 3/9J13 can be described as dm(7) = cm(7) — 1, if cm(7) > 0, and dm(7) = 0, otherwise. Recall that the equality characterizes the monoids which are complete intersections. Let A be again an arbitrary commutative ring. The monoid homomorphism 7T7 : Nn+1 ->• M is the restriction of the homomorphism (?r7) : Zn+1 ->• Z with a i—)• (a, 7). This group homomorphism gives rise to a homogeneous A-algebra homomorphism

±:L from the graded algebra A[T ] = A[T\T0...Tn of Laurent polynomials in ±:L the indeterminates TO, ..., Tn onto the graded algebra A[X ] = .Apfjx of Laurent polynomials in X. The homomorphism (TT^) = (T^)T0—Tn is obtained from TT^ simply by forming rings of fractions (Nenneraufnahme) with respect to multiplicative sets generated by TO, ..., Tn and X, respectively. In particular,

Note that for every relation (a, (3) of the monoid M the binomial is associated to the Laurent polynomial Ts — 1, where s := a—(3 G Syz(7) = ker(7T7). 4.4 Theorem Assume A ^ 0. (1) A system s(p) G ker(7r7) = Syz(7), p G P, of syzygies 0/7 generates Syz(7) if and only if the Laurent polynomials Ts -p' — 1, p G P, generate the ideal ker (TT^) = 3A[T±l}. (2) Let (a(j),(3(j)), j G J, be a minimal system of relations of M, Sj := {a(j'),7} = {/3(j')57} the degrees of the corresponding binomials Fj := Ta(-^ - T/3^ € 3 and let m € N. Then the ideal 28 I Preliminaries

is generated by Ts — 1, s G F(rri;7). Thus

Proof. For any system s(p) e Syz(7), p € P, A[T±l]/(Ts(-^-l; p € P) is n+l canonically isomorphic to the group algebra A\L / ^p£P ^s(p)] over A. From this (1) follows directly. The chain

of inclusions yields (2). 4.5 Corollary In the situation of Theorem 4.4(2), for k € N let Ek(J) denote the set of subsets E C J such that \E\ = k and that the syzygies a(j) ~ P(j)> J G E, are linearly independent. Then, for arbitrary r G N,

Proof, m G Satr(7) if and only if rankF(m;7) > n — r. Therefore the assertion follows by 4.4(2). D 4.6 Corollary In the situation of Theorem 4.4 let the monoid M = Mon(7) be a complete intersection. ThenT(rn;^) is a direct summand of Syz(7) for every m G N, and for every r G N

Proof. If M is a complete intersection, then a(j) — /3(j), j G J, is a basis of Syz(7). So the result follows from Corollary 4.5. n

Supplements

1. a) Let if be a field, f\,..., Fm arbitrary homogeneous polynomials of positive degrees <5i,..., 5m in the graded polynomial algebra if [To,..., Tn], degTi = 7i > 0, i = 0,..., n, and B the algebra K[T0,..., Tn]/(Fi,..., Fm). Then

i. e. every coefficient of the power series on the left side is greater than or equal to the corresponding coefficient of the power series on the right side. (Induction on m,.) 4 Relations of Numerical Monoids 29 b) For a numerical monoid M = Mon(7o,... , 7n), the ideal 3 = kerTrJ^ C if [To,..., Tn] is generated by at least n elements. (Let f\,..., -Fm be homo- m+1 m geneous generators of the ideal X By a) now, 1/(1 - Z) = PK(x]/(i - Z) > PK[M\/(I - Z)m > 1/(1 - Z70) • • • (1 - Z7"). This is impossible for m < n. Of course, the result is also a consequence of Krull's principal ideal theorem.) 2. t Let M be a numerical monoid generated by positive weights 7 = (70,..., 7n). In the following we use some concepts and results of Chapter II. a) The following conditions are equivalent: (1) M is a complete intersection. (2) For every noetherian ring A ^ 0 the kernel of TT^ : A[T\ —> yl[M] is generated by n binomials. (3) For every noetherian ring A ^ 0 the graded monoid algebra yl[M] is a complete intersection over A. (4) There is a field K such that the graded monoid algebra K[M\ is a complete intersection over K. b) Assume that A is a noetherian ring ^ 0 such that every finite stably free .4-module is free (which for instance is true if all finite projective ^4-modules are free or if the Krull dimension of A is < 1). Equivalent are: (1) M is a complete intersection. (5) For every surjective homogeneous ^4-algebra homomorphism the kernel is generated by m homogeneous polynomials. (Suppl. 4 of Sect. 7.) c) Assume that M is a complete intersection and that for some A ^ 0 the kernel of ?r^ : A\T\ —> yl[M] is generated by n homogeneous polynomials of degrees <5i,..., 5n • Then

M is symmetric with (See Prop. 7.2 and Suppl. 5 of Sect. 3.) Furthermore, because of C/M(!) = 1

3. Let M = Mon(7) be a numerical monoid which is a complete intersection. Then

(Let <5i,... ,5n denote the degrees of the minimal relations of Mon(7) and Ij := M + Sj . Then U^Vm = EJ=i ^ = E"=i T'sat"-^) because for any s € N the number c(s) of j with s € Ij coincides by 4.6 with the number d(s), for which s € Sat"-d(s)(7), s i Satn-d(s)-1(7).) 4. Let M be a numerical monoid generated by positive weights 7 = (70,..., 7n). a) Let 7n+i € M+ and 7 := (70,... ,7n,7n+i). Then

a If 7n+i = (0,7), the new relation is Tn+i — T . A simple example is the complete intersection N generated by 7 = (70,..., 7n) j with 70 = 1, where the ideal of relations is generated by Tj — T0 ' , j = 1,..., n. 30 I Preliminaries

Starting from the uniquely determined system of generators of M the formula 72-7 (1) = 1+7£7(1) shows in an elementary way that the deviation devM = dev7 = 72.7(1) is independent of the chosen system 7. b) Let 7 = (70, d7i,..., d7n)> 7' := (70,7i, • • •, 7n) be a situation of reduction in codimension 1 as in Section 3. Then

(If a—m, then cm(7j < 1: For every a € Nm(7J, ao f 0. If d\m, a € Nm(7J and the corresponding a' € Nm/d(7') have the same support. Thus cm(.'i) = Cm/di'j')-) c) Compute 1l~, for 7 = (24,32,40, 26,39) using a) and b). 5. Let M be a numerical monoid generated by positive weights 7 = (70, •.., 7n). a) g7 + maxi7i < con7 < g7 + min^ + max^ . (g7 +7^ (JL M + 7i 5 Con(7).) If g7 + maxi 7^ < con7 , then rmax7 = con7 . (The equality g7 + max^ 7$ = rmax7 = con7 may happen, e. g. for 7 = (3,4,5, 8) or 7 =(7,12,15, 20).) b) Let Ei be the least positive multiple of 7$ contained in Mon(7j '• j ^ i) and Ei = Vi'ji = J^ . ,. Oij7j with Vi, otij € N. Then the binomial belongs to a minimal set of binomial generators of the Ideal J of relations of M with respect to 7. In particular, Ei £ Con(7) and maxi7i < maxiUi < rmax7 < con7 . ({«} is a connected component of GE{ (7) ^ {*}•) The Ei are called the corners and the specified binomials of degree Ei the corner binomials of 7. c) Let 7 = (70, ..,7n,7n+i) with 7n+i > 0. Then equality holds if 7n+i € M. In particular, equality holds if 7n+i e M. (Let 7n+i = /3o7o H + Pn7n . If a € Nm(7) then (ao,..., an) + an+i(/30, ...,/3n)€ Nm(7).) d) Suppose 70 < 7i < 7n for all i and con7 > g7 + 7n • Then con7 = rmax7 . Furthermore, con7 = — t + 70 + 7* where t is an element of the type set T(M) and 7n + g7 — (~t) < 70 + 7fe , 7o < 7* • (For 7^ one can take any element for which k does not belong to the connected component of 0 in Gm{"f), m := con7 . For an example consider 7 = (6,8,9,11,13) where con7 = 26, g7 = 10, — t = 7.) 6. Let M be a numerical monoid generated by positive weights 7 = (70, • • •, 7n)- By <5i,..., 6r we denote the degrees of the elements of a minimal system f\,..., Fr of homogeneous generators of 3 = ker TT^- , K a field and li^ : K\F\—± K[M] the canonical homomorphism. Then rmax7 is the supremum of 61,..., 6r • ±l ±l a) An element m € N belongs to VSat(7) if and only if 3mK[T ] = 3K[T ]. (4.4(1).) From this follows b) If M is a complete intersection (i. e. r = n), then n Sat(7) = VSat(7) = P| =1(M + Sj) and sat7 = vsat7 = g7 + rmax7 .

(This follows partly from 4.6, too. The equalities sat7 = vsat7 = g7 +rmax7 may hold if M is not a complete intersection, e. g. 7 = (6, 7,8,9), but in this example 4 Relations of Numerical Monoids 31

Sat(7) = VSat(7) ^ flj=i(M + <5j), r = 4, 1. Assume that there is an element 6 € N such that Gj,(7) = {0,..., n} and that every component of Gb("i) is a single point. Then 6 = a-^i , i = 0,... ,n, with a,i € N*. The elements ao,..., an are pairwise relatively prime, 6 = ao • • • an and the ideal 3 of relations of M with respect to 7 is (minimally) generated by the binomials Fj = Tg° — Tj 3'' , j = 1,... ,n. In particular, M is a complete intersection. Its Poincare series (cf. Suppl. 2) is

(For i ^ j one has b = k • lcm(7i,7j) and necessarily k = 1, since otherwise the elements i,j would be connected. Then obviously gcd(ai,%) = 1 and 6 = OQ • • • an • Thus, M is a monoid of the type treated in Sect. 3, Suppl. 12b). To prove that 3 is generated by f\,..., Fn it is enough now to show that the M-ideal Satn^1(7) is generated by 6. Consider m € Satn^1(7). There are two different representations m = (a, 7} = (/3,7} with a, j3 € Nn+1. There is some index i with cti > ft , say i = 0. Then (ao — /3o)7o € "Z"fi + • • • + "Zi"fn = "Zdo , and there is an element c € N* such that ao = A) + cao . Thus Conversely, if Sat" (7) is a principal Mon(7)-ideal, 7 = (70,... ,7n) arbitrary, n > 1, with generator 6', then the ideal 3 of relations is generated by binomials of degree 6 . Therefore, we are necessarily in the situation discussed above with 6 = 6'. This was also proved in [17], Beispiel 5.12. Another characterization of the situation above is to say that all the relations are generated by relations of the same degree.) 8. t Let M be a numerical monoid generated by positive weights 7 = (70, •.., 7n), n > 2. Assume that there is an element 6 € N such 65(7) = {!,...,n} and that every component of Gj,(7) is a single point. Let d := gcd(7i,... ,7n). Then 6 = fl;7; , i = l,...,n, with a; € N*. The elements ai,...,an axe pairwise relatively prime and 6 = ddi • • • an . Furthermore, let EO be the least positive multiple of 70 contained in Mon(7i,... ,7n), EO = fo7o = 0171 + • • • + on7n • Then Eg ^ b and the binomials are part of a minimal system of binomial generators of J. These binomials form a full system of generators of J, i. e. M is a complete intersection, if and only if vo = d. In case n = 2 the last condition is always fulfilled. (We give only a hint for the case n = 2. Let 7^ := 7i/d, i = 1,2, and assume 70 ^ Mon(7(,72). Then there are ai, 02 € N such that

Thus 6 = d7i72 = dfjo -\- (ai + 1)71 + (02 + 1)72 , which contradicts the structure of Gj,(7). Remark: If n > 3, in general M is not a complete intersection, e. g. for 7= (11,6,10,15).) 9. t In this supplement we consider numerical monoids generated by three positive weights 7 = (70,71,72) • The structure of such monoids can be described explicitly, which was done by S.M.Johnson [20], J.Herzog [16] and J.Kraft [29]. The following approach includes their results. The ways to carry out actual com- 32 I Preliminaries putations have been studied for a long time, too, starting with observations by Johnson; for this topic see J.L. Davison [10]. Let the ground ring A be a field K. There are three partition types for a set Gm := Gm(-y) with cm(-y) > 1, m € N, which can be indicated in the following self-explanatory way: 000, •00,0 CZD • In the cases where there is an element m € N such that Gm is of type 0 0 0 or of type -00, the monoid M is a complete intersection by Suppls. 7 and 8. Therefore, we will assume from now on that the non-empty, non-connected Gm are of type 0 (• o only. a) At the corners E^jEijEz each GE^ is non-connected. In our special situation therefore these corners are pairwise different and in particular M is not a complete intersection. Fix representations Eg = i/'o7o = di7i + 0272 , E\ = Vi'ji = 6o7o + 6272 , E% = t'272 = Co7o + Ci7i and let & be the matrix

The corresponding corner binomials p rpVQ rpairpa'2 p rpvi rfjb()rpb>2 p rpv>2 rpCQrpCi 1 ° •— -'O -'l -'2 ' ri -~ -'I -'O -'2 ' ^2 -~ -I2 -'O -'l form a minimal system of generators of 3. (Otherwise there would be an element m £ {Eo,Ei,E2} such that Gm is of type 0 GZD. We may assume m = «7o = vyi + wy-z . Then « > VQ and m = (u — ^0)70 + (v + 01)71 + (w + 02)72 , a contradiction.) In particular

(In general, for any almost arithmetic sequence 7 of weights an explicit minimal set of binomial relations can be constructed, cf. Patil [32]. 7 = (70,..., 7n) is called an almost arithmetic sequence ifnof the weights form an arithmetic sequence.) b) First we develop some properties of the matrix 6, also proved by S.M. Johnson and J. Herzog: All the entries of & are non-zero. The sum of its columns is the zero-column. & is uniquely determined, i. e. the corner binomials are uniquely determined. (Assume a-2 = 0, i. e. EQ = fo7o = Qi7i • Then 0,1 > Vi because -Bo 7^ E\. There is another representation EQ = fli7i + 0272 with a^ ^ 0 because of the type of GEO • Similarly, there is a representation E\ = 6o7o + 6272 with b'0 ^ 0. Now (ai — a'i)7i = °272 and therefore ai — a'i > Vi, say ai = Vi + a'i + r. It follows that (t'o — 6o)7o = (fl'i + ^)7i + 6272 > 0 and VQ — 60 < VQ , contradiction. We have 0 < (60 + co)7o = (t'i7i — 6272) + (^'272 — Ci7i) = (vi ~ 01)71 + (i>2 — 62)72 • If Vi - ci < 0 then v-2 - 62 > 0, (v2 - 62)72 = (60 + Co)7o + (ci - i>i)7i € Mon(7o,7i) and v2 > V2 — 62 , contradiction. Hence Vi — Ci > 0, v2 — 62 > 0 and 60 + CQ > VQ . Similarly ai + c\ > v\ and 122+62 > v% . This means that all the elements of the sum of the columns of & are non-positive. Since this sum is a syzygy of (70,71,72), it is zero. Because the sum of the columns of & is zero for any choice of the elements ai, a>2, 60, 62, Co, Ci , these elements are unique.) c) In K[T] the corner binomials are (non-associated) prime polynomials. (Every element of a minimal generating system of homogeneous polynomials of the prime ideal 3 is prime.) 4 Relations of Numerical Monoids 33

The Lasker-Noether decomposition of the ideal (F(hFi) in K[T\ is

1 where the ideal (7^°, T" ) is primary to (T0,Ti), and similarly for (F(hF2), (FijFz). (Obviously, (Jo,FI) is contained in the intersection. For the converse, it remains to prove: If H • F2 € (T^°,T^) then H € (F0,Fi). But obviously To",?1™1,1*2 is a regular sequence. Therefore H € (Tg0,!1™1). Thus we have to 0 1 1 show that (Tg ,! " )!^ C (F0,Fi). But, using b) one gets simply: d) As an application of the primary decomposition in c) we get the following identity in K [T*1]:

By Theorem 4.4(1) this is equivalent with the fact that the syzygies (VQ, — a,i, —02) and (—60, Vij —62) form a basis of Syz(7), i. e. that the 2 x 2-minors of the first two columns of the matrix & are relatively prime. The analogue holds for the minors of any pair of columns of &. From that follows for the adjoint of © :

In particular

(Because (adj 6) -6 = 0, every row of adj & is a syzygy of the columns of ©. Since the components of such a row are relatively prime, every row coincides up to sign with (70,71,72)- From 6 • (adj <3) = 0 and 6 • *(1,1,1) = 0 follows that every column of adj 6 is constant. So it remains to show that one element of adj© is non-negative. But, e.g. (—ai)(—62) — Vi(—02) = 0162 + Via? > 0.) e) The ideals are generated by regular sequences. The canonical Mayer-Vietoris sequence

yields for "Put = T>K(T\/'3 the identity

with 34 I Preliminaries as numerator, where the exponents are greater than Eg, -Ei, EZ. It follows that the Frobenius number of M is

Furthermore, DI f Dz , and Di,D% are related by DI + D-2 = EO + EI + E-2 , DI • D-2 = -Eo-Ei + EoE-2 + EiEz — 707172 • (Di = 0272 + EI = Ci7i + -Bo and D-2 = Co7o + -Ei = 6272 + -Bo by b). Useful are the relations 6070 + 0171 = -Bo + -Ei — E% and 60700171 = EgEi — 707172 • Note 6001 = t-'ot'i — 72 , cf. d).) f) The degree of singularity of M is determined by the following formula of J. Kraft:

by Sect. 3. One employs the formula

which holds for any rational function of type

with mi,..., mr,ni,..., nr € Z\{0}, p € Z, and uses the hints given in e).) g) The type set of M, i.e. the set of minimal generators of QA-/ , is

(One way to prove this is by using simple direct computations similar to those used in c). We prefer to give a different proof which sheds more light on the nature of the whole set-up. Let O denote the Af-ideal (-Di + |7| + M) U (-D2 + (7! + M). We will prove that Pci = -PM(1/Z) = TaM , from which Q = OM follows. Vn is equal to the Poincare function "Pa of the graded -fiT[M]-module ui :~ 1 K[M]xi +K[M]x2 C K1X* ], where

The syzygies of Xi, X2 are generated by the columns of the matrix

That the columns are syzygies of Xi,X2, can be easily checked. Consider any c c relation HiXi +H2x2 = 0, HI, H2 e K\T}. Multiplying by Jf i+ 2-S2-|7l yields 1 1 1 a2 bo HiT" - H2Tg° € Jn (jfo/I'? ) = (F0,I\), from which HI e (T^ ,T2 ,T0 ) follows. Thus the columns of 21 generate the syzygies of Xi,X2 over K[T]. The syzygies of the columns of 21 are simply -fiT[T]-(-Fo, f\,F2): Apply Cramer's rule. Thus there is a natural exact sequence of graded K [T]-modules and 4 Relations of Numerical Monoids 35 hence "Pw = — TM(1/Z), which finishes the proof. The syzygies of FQ ,Fi,Fz are generated by the rows of 21, which are linearly independent. There is a corresponding natural exact sequence of graded K[T]-modules, which yields the (already known) formula for ~PM • In addition, this sequence can be used to compute the dualizing module UK[M] , which is isomorphic to Ext^j^-K"[M],K[T](—17|)) and hence isomorphic to the residue class module of K[T](Di — |7J) 0 K[T](£>2 — |7|) modulo the columns of 21. Thus u> = UK[M] 1S indeed a model for the dualizing module of the Macaulay ring K[M\, with corresponding M-ideal OM-) {Remark. The formulas for gM and I?M hold also in the case of a complete intersection generated by 70,71,72 • This is easily verified using the explicit descriptions in Suppls. 7,8. Note that also in this case the numerator of TM can be Dl D 2 written in the form 1 — 72.7+Z -\-Z ' with well-defined different positive integers Di,D2. Assume DI < D2. Then D2 = gM + J7J = E0 + EI and 707172 = E0Ei if Eg = E'2 . The type set of M is simply {—D-2 + |7|}. The element —Di + 7) belongs here to QM\T(M).) 5 Splitting of Numerical Monoids

A useful tool in the handling of numerical monoids is a splitting procedure, first considered by K. Watanabe and Ch.Delorme, cf. [11]. As in the last sections we consider a numerical monoid M = Mon(7) generated by positive weights 7 = (701 • • • > 7«)- Let m G N and let us assume that 0 := {0,..., n} is decomposed into two disjoint sets: 0 = fi'ttJfi" such that both fi' n Gm(7) and 0" n Gm(7) are non-empty unions of connected components of Gm (7); for the definition of Gm (7) see the remarks following Proposition 4.3. Furthermore, define a' := gcd(7j : i G 0'), 7' := (7i/a')jeo>, M' := Mon(V), r' := Syz(V) l and similarly a", 7", M", T" with respect to fl". Then M = a'M' + a"M", m = a'm' = a"m" with m' G M', m" € M" and m = o'o"d with d = gcd(m', m"). 5.1 Proposition In t/ie situation just described, there is a canonical exact sequence of abelian groups

In particular, m G Sat(7) if and only if m' G Sat(7') ond rn" G Sat(7"). In that case

Furthermore, m € VSat(7) if and only if d = 1 and m' G VSat(7'), m" G VSat(7"). Proof. We will use the canonical decomposition Z = Z ®Z . Choosing elements a' G Nm/(7'), a" € Nm//(7") one has where a := (a1, -a") = a' - a" e Zn = Zn/ ® Zn". The (surjective) linear form 7 : Z —)• Z is the composition

From the canonical commutative diagram

1 Recall that we sometimes use the abbreviation F = 872(7).

36 5 Splitting of Numerical Monoids 37 with exact rows one gets the desired exact sequence of the Proposition as the exact sequence of the cokernels, because the image of a in Z(o", —a') is the element (m', -m") = d(a", -a'). a Let us say that M = Mon(7) splits along fl', fl" if where a' & M" = Mon(7") and a" & M' = Mon(V). Then we are in the situation studied above for m = a'a", d = 1, m' = a", m" = a', because the support of any a G JMa'a"(7) necessarily is contained in il' or in fl". In this case the monoid M is canonically isomorphic to the quotient (M1 (&M")/R, where R is the compatible equivalence relation generated by the single relation ((m',0), (0,m")) = ((o",0), (0,o')). Proof. Under the surjective homomorphism

by (#,3/) <—> a'x + a"y two elements (#1,3/1), (#2,3/2) have the same image if and only if (#1,3/1) = (#2,3/1) + k(m',0), (#2,3/2) = (#2,3/1) + k(0,m") or (#i,3/i) = (#i,3/2) + A;(0,m"), (#2,3/2) = (#1,3/2)+A;(m',0) for some A; e N. To put it in a different way: M is isomorphic to (a'M1 ® a"M")/R, where .R is generated by ((m, 0), (0,m)), m = a'a". This description of M shows that there is a canonical isomorphism

of graded Jf-algebras. It follows: 5.2 Theorem If M splits in the way described above, then

a a Proof. The formula for PM follows from the facts, that TM'(Z ')TM"(Z ") is the Poincare series of the tensor product and that Xm 1 — 1 Xm is a non-zero-divisor (because the tensor product is an integral domain contained in K[o'N] ®K K[a"N]). Multiplying by 1 - Z yields

A comparison of degrees proves the formula for gM • The formula for -&M follows easily from '&M = G'M(1.) = Q'M(\)/QM(^)I using logarithmic deriva- tives. D 38 I Preliminaries

The formulas for gM and "&M in Theorem 5.2 imply

From this follows directly, that the splitting monoid M is symmetric if and only if its components M' and M" are both symmetric. This application as well as the formula for gM were already proved in [11], Proposition 10 in a direct way. More generally, the representation of .K"[Af] from which Theorem 5.2 was derived allows to prove that the type of M is the product of the types of M' and M", i. e. IM = ^M'^M" • This can also be proved directly in a straightforward way. Coming back to the aforementioned representation of Jf[M] we note first, that the kernel 5' of K[Ti : i € fl'] ->• K[a'M'] is the kernel 3' of K[T'} ->• K[M'}, subjected to the isomorphism K[T'} ^ K[Ti : i e 0'] by T[ i—> Ti (which multiplies degrees by a'). Analogously, the kernel 3" of K[Ti : i e fl"} -)• K[a"M"} can be identified with the kernel 3" of K[T"} ->• K[M"}. Thus the kernel 3 of K[T] ->• K[M] is generated by 5', a 5" and the element F := T '® 1- 1®T«", where a' € Nm/(V) = Nm(o'V) and a" G Nm//(7") = Nm(o"7") are fixed. Obviously, minimal systems of binomial generators of 3' and 3" together with F make up a minimal system of binomial generators of X In particular: 5.3 Proposition In the situation just described,

Recall that dev7 = 7£7(1) — n. As a corollary one gets that the splitting monoid M is a complete intersection if and only if both its parts M' and M" are complete intersections. This was also proved in [11], using slightly different methods.

Supplements 1. Let M be a numerical monoid generated by positive weights 7 = (70,..., 7n)- a) If rmax7 € VSat('y), then sat7 = vsat7 = g7 + rmax7 . (One is in the situation of 5.1 with d = 1 and rmax7 = a'a". Consider I := g7 + rmax7 € M. There is a representation I = a'x' + a"x" with x' € M', x" € M", which is easily seen to be unique. From this follows

In particular rankF(I;7) < n. Thus I ^ Sat(7) and I < sat7 < vsat7 . Equality now follows by Suppl. 6a) in Sect. 4. Examples where rmax7 € VSat(7) and M is not a complete intersection are provided by 7 = (24,32, 40, 26,39) and 7 = (33,44,55, 26,39).) b) Suppose M ^ N. Then rmax7 < con7 < vsat7 . 5 Splitting of Numerical Monoids 39

(If con7 > vsat7 then rmax7 = con7 . Thus it suffices to show rmax7 < vsat7 . Assume the opposite. Then vsat7 < rmax7 = vsat7 — g7 by a), hence g7 = —1, 1. e. M = N. Contradiction. For M = N one has rmax7 = con7 = vsat7 + 1.) c) Suppose M ^ N. If rmax7 > sat7 or con7 > sat7 , then sat7 < rmax7 = con7 < vsat7. (The hypothesis implies rmax7 = con7 > sat7 . The inequality con7 < vsat7 follows by b). An example of 7 with rmax7 > sat7 is (6,7,9,17). In this case sat7 = 32 and rmax7 = con7 = vsat7 = 34. Other examples are given by 7 = (6,14, 21,73) with 73 = 43 or 73 = 37. In these cases sat7 , rmax7 = con7 , vsat7 are 80,86, 86 and 73, 74,80 resp.) d) The following conditions on 7 are equivalent: (1) con7 e VSat(7). (2) M = Mon(7) is symmetric and con7 > g7 + max; 7; . (3) M = N or M is a complete intersection of the following special type: There are a decomposition {0,..., n} = O ttl O and a , a € N, 7 = (7;);gn' , 7 = (7")isn» such that 7 = (aYXY')i max; 7; < a' + a" and M = Mon(a',a")- (The implications (3) => (1) and (2) =^ (1) follow by direct computations. For the other implications we may assume M ^ N. For the proof of (1) =>• (3) note that con7 = rmax7 . Thus we may use a) and the decomposition as in 5.1. We have con7 = a'a" > g7 + max; 7;. Because of a'gy + a"gy> + a'a" £ M one has g7 > a'gy + a"gy> + a'a" and therefore 0 > a'gy + a"gy> + max; 7; , which implies gy = gy/ = —1. For the proof of (2) => (3) one uses Suppl. 5d) in Sect. 4 and assumes 70 < 7; < 7n • One has m := con7 = g7 + 70 + 7* and 7n < 7o + 7fe where k is any element of {0,... ,n} not belonging to the connected component of 0 in Gm(-y). Therefore it suffices to show that every 7; ^ 7^ is a multiple of 70 . Using an induction argument it suffices to show 7^—70 G M. But, if 7; — 70 ^ M, there is an element x € M with 7;—7o+£ = g7 = w—70 —7* and m = 7i+7fe+x. Contradiction.) 2. Assume that the numerical monoid M = Mon(7) = a'M' + a"M" splits along O', O", where fi' ttl O" = {0, ...,n}. We will use the corresponding notations introduced in this section. Let Vi, i = 0,..., a — 1, denote the minimal elements of M' with v-i = i modulo a" and Wj , j = 0,..., a' — 1, the minimal elements of M" with Wj = j modulo a'. Then are the minimal representatives in M for the residue classes modulo m = a'a". This provides a simple proof of Theorem 5.2. 3. t Assume that the numerical monoid M = Mon(7) = a'M' -\-a"M" splits along O', O", where O'ttlO" = {0,..., n}. We will use the corresponding notations introduced before. a) As in the procedure of reduction in codimension 1 (cf. Section 3) there are canonical reduction maps R' : Z —y Z and R" : Z —>• Z which we define as follows: Let m € Z. There is a unique minimal element s'(rn) € M" such that m = a"s'(m) modulo a' and a unique minimal element s"(m) € M' such that m = a s (TO) modulo a . Then 40 I Preliminaries

One proves readily (Analogous results hold for R".): (1) TO € M if and only if R'(m) € M'. (2) R'(m + a'z) = R'(m) + z for allTO , z € Z. (3) If I is a fractional M-ideal, then R'(I) is a fractional M '-ideal. (4) If /' is a fractional M'-ideal, then I := R'~l(I') is a fractional M-ideal with Frobenius number g/ = a g/' + a (gA-f + a ). n+1 n/ n b) Let us fix a' € Na//(V), a" € Na/(y) and identify Z = Z 0 Z ". Then

There is a natural system of generators for Syz(7) respecting this decomposition: Let Fj := Ta<-^ — Tt3

If TO € M + a'a", then

As a consequence, ifT O ^ M + a'a" then SyZ(7)/r(m; 7) = SyZ(7')/r(E'(m); 7') 0 SyZ(7")/r(E"(m); 7") 0 Z(a', -a"), and ifT O € M + a'a" then d) From c) follows:

The Frobenius numbers sat7 and vsat7 are therefore the maxima of the Frobenius numbers of the Af-ideals on the right side of the equations above, which can be written down using a) (4). (Examples show that none of the ideals in the intersection formulas can be neglected, for instance, consider 7' = (3,8), 7" = (5,7) and (a', a") in the three cases (5,3), (5,6), (36,25).) References

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