University of Paderborn Department of Mathematics. Diploma Thesis. On stable monomial ideals

Transcription

1 University of Paderborn Department of Mathematics Diploma Thesis On stable monomial ideals by Kai Frederik Gehrs Paderborn, January 27, 2003 Thesis Advisor: Prof. Dr. Uwe Nagel

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3 Erklärung Ich versichere, dass ich die vorliegende schriftliche Arbeit selbständig angefertigt und keine anderen als die angegebenen Hilfsmittel benutzt habe. Alle Stellen, die dem Wortlaut oder dem Sinn nach anderen Werken entnommen sind, habe ich unter genauer Angabe der Quelle deutlich als Entlehnung kenntlich gemacht. Paderborn, den 28. Januar 2003, Kai Frederik Gehrs

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5 Im großen Garten der Geometrie kann sich jeder nach seinem Geschmack einen Strauß pflücken. David Hilbert ( )

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7 Acknowledgement I would like to thank Uwe Nagel for making this work possible, even though he had to do so many other things within the last months Benno Fuchssteiner, Oliver Kluge, Walter Oevel, Andreas Sorgatz and all the other people in the MuPAD Research Group for teaching me programming in MuPAD and including me into their group Thilo Pruschke for proofreading a great part of my work and helping me, when I had questions during the whole course of my study Christopher Creutzig for introducing me to LaTeX Jörn Maas for his technical support Susanne Happ, Petra Neumann and Hubertus R. Drobner at Studienstiftung des Deutschen Volkes for their support and encouragement my family Bodo, Dodo, Michael, Julia and Jenni Gehrs and my friends James Hetfield for the wonderful music

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9 Abstract The main result of this thesis is to present an algorithm for the computation of all saturated stable ideals to a given Hilbert polynomial. It uses and adapts ideas of the Ph.D. thesis Combinatorial Structure on the Hilbert Scheme by Alyson Reeves (see [16]). The class of stable ideals is introduced, and some useful properties of these ideals are discussed in detail concerning computation of Hilbert series, Hilbert polynomials and saturation. For each of these explicit formulas are given. Special lexicographic ideals are defined, which can be associated to a given Hilbert series and a given Hilbert polynomial. These ideals play a major role in the computation of all saturated stable ideals to a given Hilbert polynomial. Apart from the theoretical results of this thesis, the last chapter presents the source code of an implementation of the algorithm to compute all saturated stable ideals to a given Hilbert polynomial. Additionally, the source code of various other algorithms concerning stable ideals, e.g. to compute Hilbert polynomials, Hilbert series or Hilbert functions is discussed in Chapter 4. As a basis for the implementations, the computer algebra system MuPAD is used. The algorithms can be run on any available version of MuPAD and do not make use of high-level library functions. Therefore the code can easily be implemented in any other comparable computer algebra system.

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11 Contents Introduction 3 1 Notations and Prerequisites General prerequisites and terminology Hilbert function, Hilbert polynomial and Hilbert series Characterization of Hilbert functions Stable ideals Borel-fixed ideals Stability of Borel-fixed ideals Saturation of stable ideals Hilbert series of stable ideals A link between Hilbert series and stable ideals A link between Hilbert polynomials and stable ideals Hilbert polynomials of stable ideals An application to Gotzmann s Regularity Theorem Operations on stable ideals Left-shifts and right-shifts of monomials Expansions and contractions of monomials Stable ideals with the same double saturation Stable ideals with the same Hilbert polynomial Algorithms for stable ideals Computing Hilbert series of stable ideals Computing Hilbert polynomials of stable ideals Computing the lexicographic ideal L p Computing expansions and contractions Computing stable ideals to a given Hilbert polynomial Computing all Hilbert series to a Hilbert polynomial Computing all Hilbert functions to a Hilbert polynomial Some conclusions and experimental results Glossary of Notations 127 1

12 Glossary of MuPAD procedures 129 Bibliography 131 2

13 Introduction The main result of this thesis is to present an algorithm to compute all saturated stable ideals to a given Hilbert polynomial according to the Ph.D. thesis Combinatorial Structure on the Hilbert Scheme by Alyson Reeves (see [16]). The importance of the study of stable ideals to the topic of algebraic geometry has been recognized through the work of several researchers. The contributions of Robin Hartshorne (see [8], [9]), David Bayer (see [1]) and Mark Green (see [4], [5]) are of particular interest. Within their work, a special class of monomial ideals the so-called generic initial ideals has been studied. The reader, who is not familiar with the definition of the generic initial ideal to a given homogeneous ideal I of the polynomial ring K[x 0,..., x n ] in n+1 variables over the field K may think of it as an initial ideal (i.e. a monomial ideal, generated by the leading monomials of all homogeneous elements of I with respect to some monomial order), which can be obtained from I after the application of a sufficiently general change of coordinates to its generators. A theorem proved by Galligo, Bayer and Stillman, which can be found for example in the book Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud (see [6], Chapter 15, Theorem ), states that if I is a homogeneous ideal in the polynomial ring K[x 0,..., x n ] in n + 1 variables over the field K, the generic initial ideal of I will be Borel-fixed. Within this thesis, the field K will always be of characteristic zero. In this case, Borel-fixed ideals are stable ideals, i.e. they are characterized by a certain combinatorial property. Generic initial ideals also played an important role in Hartshorne s proof of the connectedness of the Hilbert scheme. The Hilbert scheme is an interesting and rather complicated object studied in algebraic geometry. With the help of this scheme, one can parametrize the subschemes of the projective space P n with the same Hilbert polynomial. It consists of different components, whose number can be estimated by the number of stable ideals with the same Hilbert polynomial. Hence, for further observations of the Hilbert scheme it would be useful to have an algorithm to compute all such stable ideals. Thus, the reader might see that it is indeed of interest to study the class of stable ideals not only from the point of view of commutative algebra, but also from that of algebraic geometry. 3

14 This thesis is organized in four chapters: Chapter 1. Notations and Prerequisites In order to enable not only readers with a good background in commutative algebra and algebraic geometry to follow the train of thought presented here, the first chapter of this thesis sets the stage: Basic terminology will be introduced to the reader and basic definitions will be recalled. Some well known results concerning primary decomposition of ideals, saturation of ideals and Hilbert polynomials will be quoted. Another important result at the end of the first chapter is the characterization of the Hilbert function of a homogeneous K-algebra, which goes back to F. S. Macaulay and will not be proved here, either. We will make use of this result, when we prove the correctness of an algorithm stated in Chapter 2. All results, which are stated and not proved at all, can be accepted without the knowledge of any details of their proof, since we will not make use of these details later. Chapter 2. Stable ideals In the second chapter, we define Borel-fixed ideals. Under special conditions, we characterize these ideals by a certain property (due to [6]), which leads us to the class of stable ideals. These ideals will be of main interest in the following chapters. The theory to compute the saturation, the Hilbert series or the Hilbert polynomial of a stable ideal is presented. To compute each of these, we give efficient algorithms. Furthermore we establish a link between Hilbert series and stable ideals, i.e. we will describe an algorithm to compute a unique saturated lexicographic ideal to a given Hilbert series of K[x 0,..., x n ]/I for a saturated homogeneous ideal I K[x 0,..., x n ]. We will see that this ideal is also stable. Similarly, we describe how to compute a unique saturated lexicographic ideal to a given Hilbert polynomial, which is again a stable ideal. As an appendix of Chapter 2, we deal with an application of the former results to a consequence of Gotzmann s Regularity Theorem (see [5], Chapter 3 ). In particular, we point out that Gotzmann s upper bound on the degrees of the minimal generators of ideals to a given Hilbert polynomial cannot be improved. Chapter 3. Operations on stable ideals Chapter three is the theoretical core of this thesis. For stable ideals, we define leftshifts and right-shifts of monomials. We will explain, what has to be understood under expansions and contractions of monomials. The reader might think of these constructions as algorithmic tools, which will be used to compute all saturated stable ideals to a given Hilbert polynomial. Later within this chapter, we will be able to 4

15 prove that all saturated stable ideals with the same double saturation and the same Hilbert polynomial are linked by a finite sequence of paired contractions and expansions of monomials. This provides an algorithmic classification of all such ideals. Additionally, we prove that under special conditions, the unique saturated lexicographic ideals to a given Hilbert polynomial and a given Hilbert series defined in Chapter 2 have the same double saturation. From these results, we will compute all Hilbert series associated to a given Hilbert polynomial p(z) of K[x 0,..., x n ]/I, where I K[x 0,..., x n ] is a saturated stable ideal. In the last section of the chapter we present a pseudo code version of the algorithm to compute all saturated stable ideals to a given Hilbert polynomial. The above mentioned contractions and expansions of monomials will play an important role within this algorithm. Chapter 4. Algorithms for stable ideals The aim of Chapter 4 is mainly to use the results presented in the preceding three chapters to establish algorithms for stable ideals. These algorithms will not only be stated in pseudo code, as for example in Chapter 2 and Chapter 3, but also in source code. Here, the MuPAD programming language serves as a basis for the source code. The code listed can always be copied and included into any of the usual versions of the computer algebra system MuPAD, which are available today. This code uses little of the high level library functions of MuPAD. Since the MuPAD programming language in its simplest form (the only form, in which it will be used within this thesis) is quite similar to the original Pascal programming language, it will be easy for anyone to understand the code who is used to Pascal or even any other of the programming languages of the modern computer algebra systems available today. Hence, it is also easy to implement the algorithms in any other of the above mentioned programming environments. In detail, we will present the source code for algorithms to compute Hilbert series of stable ideals, Hilbert polynomials of stable ideals, special lexicographic ideals, expansions and contractions and many more. Combining all these algorithms, we will see that we can compute all saturated stable ideals to a given Hilbert polynomial. The chapter ends with an appendix presenting some experimental results and conclusions concerning the algorithm to compute all saturated stable ideals with the same Hilbert Hilbert and the structure of the unique lexicographic ideals associated to a Hilbert polynomial. To check the implementations in MuPAD, several tests have been written. The testfile containing the tests is not stated within the text, because it would overstep the framework of this thesis (it consists of more than 1200 lines of code, which are useful, to check the correctness of the algorithms in practice, but not of interest within the mathematical context of this thesis). Nevertheless, the tests are available in electronic form on the CD enclosed to this thesis. On the CD one finds the complete implementation of all algorithms 5

16 6 described in Chapter 4. Additionally, with the friendly permission of the MuPAD-Research-Group and the company Sciface Software GmbH, several versions of MuPAD could be added to the CD. The user may find a MuPAD-Light version for the use of MuPAD under Microsoft Windows and a MuPAD-Light version for the use under Unix respectively Linux. They can be used for free by anyone, who is working in an academic area, and contain the whole mathematical functionality of MuPAD (the only disadvantages towards the official versions of MuPAD are the lack of graphics and the lack of a user interface). For details, we refer to the web sites and There one finds additional information on technical support and the latest versions of the MuPAD documentation files in HTML-format and in PDF-format.

17 Chapter 1 Notations and Prerequisites We want to recall some basic definitions and, according to this thesis, some central results. The more standard definitions and results will be mentioned within the text and will not be numerated explicitly (there is a quick reference of notations in the Glossary of Notations, see page 127). Definitions and results, which may not be known in general and which serve to develop the main results of this special thesis, will be stated in the usual, formal way and be numerated to make it easy to refer to them. 1.1 General prerequisites and terminology Let K be a field of characteristic zero, and let R := K[x 0,..., x n ] denote the polynomial ring in n + 1-Variables x 0,..., x n. Each element f R can be uniquely written in the form k f = r i m i, where r i K and m i = x a i0 0 x a i x a in n R with a i0, a i1,..., a in N 0. i=0 For each 0 i k we call r i m i a term of the polynomial f, r i the coefficient of the term n and m i a monomial. The degree of a monomial m = x a x an n R is deg m := a i. The degree of a polynomial f = k r i m i R is defined to be the maximum of the degrees i=0 of its monomial components m i, that is deg f := max{deg m i i = 0,..., k}. We call f a homogeneous polynomial or homogeneous of degree d, if there is d N 0 such that all terms of f have degree d. Thus, the polynomial ring R becomes a graded ring over K, graded by the degree. In each degree d, the set of all polynomials of degree d forms a vector space over K. In some situations, in particular in connection with exact sequences, we will have to shift the grading of R. For k N we write R( k) to denote that every element f R, deg f = d, has degree d k when viewed as an element of R( k). We will denote the set of all monomials of R by M, i.e. M := {m m R, m is a monomial}. 7 i=0

18 8 CHAPTER 1. NOTATIONS AND PREREQUISITES On the set M we define the lexicographic order >:=> lex. For two monomials m = x a x an 0, n = x b x an n R, we have m > n if and only if the first non-zero entry in the vector (a 0 b 0,..., a n b n ) is positive. The lexicographic order is a total order on the set M of all monomials of R. The set M serves as a basis of R viewed as vector space over K. By [R] d for a d N 0 we denote all homogeneous elements of R having degree d, i.e. [R] d := {f f R, deg f = d, f homogeneous}. For each d, [R] d itself is a vector space over K( with) a basis given by all monomials in M n + d having degree d. Note that there are exactly of these monomials having degree d n and that this implies ( ) n + d dim K [R] d =, n where dim K denotes the vector space dimension over K. This result will be useful to determine the Hilbert function, the Hilbert polynomial and the Hilbert series of R, as we will see later. In this terminology we can write R as the direct sum R = d Z[R] d or, if we shift the grading of R by k, R( k) = k Z [R( k)] d = k Z[R] d k. For an ideal I R generated by the polynomials f 1,..., f s R we write I = (f 1,..., f s ). According to the notation mentioned above, we denote by [I] d the set of all homogeneous elements of I having degree d. For each d the set [I] d and the ideal I itself can again be viewed as a vector space over K. All ideals treated in this thesis will be homogeneous, i.e. we can and will consider the elements f 1,..., f s generating I to be homogeneous polynomials. The class of ideals, which is of main interest in our context, is the class of monomial ideals. An ideal I = (m 1,..., m s ) R is called a monomial ideal if it can be generated by monomials m 1,..., m s, i.e. m i = x a i0 0 x a i x a in n, where a i0, a i1,..., a in N 0. The set of monomials {m 1,..., m s } is called a set of monomial generators of I. We call {m 1,..., m s } a set of minimal monomial generators of I if it consists of a minimal number of generators of I, i.e. if for any other set of monomials {n 1,..., n t } R generating I, we have s t. In this case, µ(i) := s is the well-defined number of minimal generators of I. The class of monomial ideals is closed under the usual operations: If I, J R are monomial ideals, then I J, I + J and I : J = {f f R, f m I for every m J} are monomial ideals. Furthermore, if I R is a monomial ideal, the primary ideals q i and the

19 1.1. GENERAL PREREQUISITES AND TERMINOLOGY 9 associated prime ideals p i, i = 1,..., r, in a primary decomposition I = q 1... q r of I may be taken as monomial ideals. For any ideal I R, we denote the radical of I to be the ideal Rad I := {f f R, there is a k N such that f k I}. If I is a monomial ideal, then Rad I is again a monomial ideal. A fact, we will use later, is that we can compute the ideal quotient of two monomial ideals quite easily: Lemma 1.1. (Ideal quotient of monomial ideals) Let I = (m 1,..., m r ), J = (n 1,..., n s ) R be monomial ideals. Then the ideal I : J can be computed as the intersection of the monomial ideals ( m 1 gcd(m 1, n 1 ),..., ) ( m r... gcd(m r, n 1 ) m 1 gcd(m 1, n s ),..., ) m r, gcd(m r, n s ) where gcd(m i, n j ) denotes the greatest common divisor of m i and n j, 1 i r, 1 j s. In the special case, where s = 1, we obtain ( ) m 1 I : J = gcd(m 1, n 1 ),..., m r. gcd(m r, n 1 ) We want to define lexicographic ideals and lexicographic subsets of [R] d for d Z. Definition 1.2. Let M R be a set of monomials of degree d Z. M is called a lexicographic subset of [R] d, if there is an r N, such that M is generated by the r greatest monomials (with respect to the lexicographic order on M) of [R] d. Definition 1.3. A monomial ideal I R is called a lexicographic ideal, if for each d Z the vector space [I] d, i.e. the vector space generated by all monomials of degree d in I, is a lexicographic subset of [R] d. For example, if R := K[x 0, x 1, x 2 ], then the set M := {x 2 0, x 0 x 1, x 0 x 2 } R is a lexicographic subset of [R] 2, because the following six monomials x 2 0, x 0 x 1, x 0 x 2, x 2 1, x 1 x 2, x 2 2 are the only monomials of degree 2 in [R] 2 and we have (with respect to the lexicographic order) x 2 0 > x 0 x 1 > x 0 x 2 > x 2 1 > x 1 x 2 > x 2 2. Thus, I := (x 2 0, x 0 x 1, x 0 x 2 ) is generated by the three greatest monomials in [R] 2. Furthermore, I is a lexicographic ideal. We will come back to these definitions in Chapter 2, where we prove that there is a unique lexicographic ideal to a given Hilbert series respectively Hilbert polynomial. In the following, we will state some basic definitions in algebraic geometry resp. commutative algebra: We need the notion of a saturated ideal and the saturation of a homogeneous ideal I R.

20 10 CHAPTER 1. NOTATIONS AND PREREQUISITES Definition 1.4. A homogeneous ideal I R is called a saturated ideal if the maximal homogeneous ideal m = (x 0, x 1,..., x n ) R is not an associated prime ideal of I. This means that if I = q 1... q s is a primary decomposition of I, we have Rad q i m for each 1 i s. If J R, J R, is not a saturated ideal, a primary decomposition J = q 1... q k q T always contains a primary ideal q T with Rad q T = m. Thus, the intersection of a finite number of saturated ideals will always give a saturated ideal. Lemma and Definition 1.5. The saturation sat xn (I) of the homogeneous ideal I R with Rad I m is defined to be the smallest saturated ideal containing I, i.e. sat xn (I) := J. J saturated, J I The existence of the ideal sat xn (I) to a given homogeneous ideal I can easily be seen: If I is saturated, we have I = sat xn (I) and there is nothing to prove. On the other hand, if I is not saturated and dim K R/I is not finite, we have a primary decomposition of I of the form I = q 1... q s q T ( ) with Rad q i m for each 1 i s and Rad q T = m. Thus, by omitting the primary component q T, we obtain an ideal J := q 1... q s containing the ideal I, which is a saturated ideal, since the above intersection ( ) is not the empty set. J is the smallest saturated ideal containing I by construction. Thus, we have J = sat xn (I) and we are done. If dim K R/I is finite, then sat xn (I) = R. In this case, I defines an empty scheme. This case is not of interest in this thesis. For further information on basic ideal theory and some detailed treatment of primary decomposition of homogeneous ideals in the polynomial ring, we refer to the book Ideals, Varieties and Algorithms An Introduction to Computational Algebraic Geometry and Commutative Algebra by D. Cox, J. Little and D. O Shea (see [3]). In Chapter 4, pp , of this book, one can find the so-called Algebra-Geometry Dictionary, which gives a compact overview on the above mentioned topics. Additional information is contained in Einführung in die kommutative Algebra und algebraische Geometrie by E. Kunz (see [11]). 1.2 Hilbert function, Hilbert polynomial and Hilbert series In this section, we introduce Hilbert functions, Hilbert polynomials and Hilbert series. Some elementary results will be quoted and not be proved here. They will become useful in later course of this thesis. Definition 1.6. For a homogeneous ideal I R the Hilbert function h I : Z Z is defined by h I (j) := dim K [I] j

21 1.2. HILBERT FUNCTION, HILBERT POLYNOMIAL AND HILBERT SERIES 11 for all j Z. Similarly, the Hilbert function of the ring R/I is the function h R/I : Z Z with h R/I (j) := dim K [R/I] j for all j Z. Here, we denote by dim K [I] j respectively dim K [R/I] j the vector space dimension of [I] j respectively [R/I] j over the field K. We will have a look at an example later on. The following result is important and goes back to Hilbert and Serre: Theorem 1.7. (Hilbert and Serre) Given a homogeneous ideal I R with Hilbert function h R/I : Z Z there is a univariate polynomial p R/I (z) Q[z], such that p R/I (j) = h R/I (j) for j 0. A proof of the theorem is given in [8], Theorem 7.5, pp The analogous result also provides the existence of a polynomial p I (z) Q[z] with p I (j) = h I (j) for all j 0. Definition 1.8. The above polynomials p R/I (z) and p I (z) are called the Hilbert polynomials of R/I and I, respectively. We say that the Hilbert function h R/I respectively h I is associated to the Hilbert polynomial p R/I (z) respectively p I (z). We introduce the latter expression, since in most cases, one can find many different Hilbert functions, which all equal the same Hilbert polynomial in large degrees. These Hilbert functions are exactly those, which we call associated to a certain Hilbert polynomial. Definition 1.9. The Hilbert series of I is the formal power series H I (t) := h I (i) t i. i Z Analogously, we define H R/I (t) := h R/I (i) t i to be the Hilbert series of the ring R/I. i Z The Hilbert series can be written as a rational expression H(t) = g(t) (1 t) n+1, where g(t) is a polynomial in t with coefficients in Z. Since the factor (1 t) or a higher power of (1 t) may divide the numerator g(t) of H(t), we may write H(t) in the form H(t) = g(t) where g(t) is no longer divisible by (1 t), i.e. g(1) 0. This representation of the Hilbert series of a homogeneous ideal I or the ring R/I is called a reduced (1 t) e Hilbert series of I respectively R/I. It is quite easy to compute the reduced Hilbert series of given Hilbert series, because we only have to check whether the numerator of the Hilbert series vanishes for t = 0. If this is the case we compute the polynomial given by the numerator divided by (1 t). Then

22 12 CHAPTER 1. NOTATIONS AND PREREQUISITES we check again: If this new polynomial still vanishes for t = 0, we repeat the division step. Otherwise, we have found the reduced Hilbert series and we are done. Furthermore, if I is a homogeneous ideal of R and H R/I (t) = g(t) the reduced Hilbert (1 t) e series of R/I, D := e 1 is the dimension of the projective scheme defined by R/I. Some very useful formulas for Hilbert functions, Hilbert polynomials and Hilbert series of a polynomial ring are summarized in the following lemma. We will make use of these results in the second chapter, where we deal with stable ideals. Lemma Let I R be a homogeneous ideal and j Z. (i) h R/I (j) = h R (j) h I (j) (ii) p R/I (z) = p R (z) p I (z) ( ) n + j (iii) h R (j) = n ( ) n + z (iv) p R (z) = n (v) H R (t) = 1 (1 t) n+1 Proof. (i) and (ii) follow from the short exact sequence 0 I ι R π R/I 0, where ι is the embedding of I into the graded module R and π is the canonical epimorphism. Recall that the vector space dimension is additive on exact sequences, i.e. dim K [R/I] j = dim K [R] j dim K [I] j for all j Z. This yields h R/I (j) = h R (j) h I (j) and thus ( p R/I ) (z) = p R (z) p I (z). n + j (iii) and (iv) follow from the fact that there are exactly monomials of degree j in n R. (v) is be proved by induction on n. In the case n = 0 we have R = K[x 0 ] and H R (t) = i Z h R (i) t i = i=0 ( ) i t i = 0 i=0 t i = 1 1 t,

23 1.2. HILBERT FUNCTION, HILBERT POLYNOMIAL AND HILBERT SERIES 13 which proves the claim for n = 0. Let n > 0 and R := K[x 0,..., x n+1 ]. By induction, we know the Hilbert series of the ring K[x 0,..., x n ]. Thus, we may write 1 (1 t) = 1 n+2 (1 t) 1 n+1 1 t = ( ) ( ) n + i t i t j = ( i ( ) ) n + j t i n n i Z j Z i Z j=0 = ( ) n i t i = h R (i) t i = H R (t). n + 1 i Z i Z Remark (Representation of Hilbert polynomials) For a given Hilbert polynomial r p R/I (z) = λ i z i of R/I, we find a special representation as a sum of binomial coefficients. i=0 There are integers m 0 m 1 m 2... m r 0 with r = deg p R/I (z), such that we can write p R/I in the form p R/I (z) = g(m 0,..., m r ; z) := r i=0 [( ) z + i i + 1 ( z + i mi i + 1 )]. This representation is found as follows: The identity ( ) ( ) z + r z + r mr = m r z r + terms of lower degree ( ) r + 1 r + 1 r! enables us to obtain the value of m r by comparing the coefficient of z r in p R/I (z) = r λ i z i and the coefficient m r of z r in ( ), which provides m r = r! λ r. To determine the value of r! m r 1 we consider the polynomial ( ) ( ) z + r z + r mr p(z) := p R/I (z) +. r + 1 r + 1 Since deg p(z) = deg p R/I (z) 1 = r 1, it follows by the same arguments as above that m r 1 = (r 1)! λ r 1, where λ r 1 is the leading coefficient of p(z). Continuing this process gives the values of all integers m 0, m 1,..., m r. Example We consider two examples: First, the case of a linear Hilbert polynomial and then the case of a Hilbert polynomial of degree two: i=0

24 14 CHAPTER 1. NOTATIONS AND PREREQUISITES (i) Let p(z) = 2z+1 be the given Hilbert polynomial. Now we proceed as in Remark 1.11 to compute m 0 and m 1 : In the terminology above we get ( r = 1) and( thus m 1 = ) 1! 2 = z + 1 z Then we compute the polynomial p(z) := p(z) + = Hence, it follows m 0 = 0! 2 = 2. Thus, we write p(z) = 2z + 1 in the form ( ) ( ) ( ) ( ) z + 1 z 1 z z 2 p(z) = (ii) Consider the Hilbert polynomial p(z) = 2z 2 + 2z + 1. In the terminology of Remark 1.11, we have r = 2. Following ( the above ) description ( ) provides m 2 = 2! 2 = 4. z + 2 z 2 Hence, we obtain p(z) := p(z) + = 6z 3. Next, we compute 3 ( ) 3 ( ) z + 1 z 5 m 1 = 1! 6 = 6 and set p(z) := p(z) + = 12. Finally, it follows 2 2 m 0 = 12 and we write p(z) as ( ) ( ) ( ) ( ) ( ) ( ) z + 2 z 2 z + 1 z 5 z z 12 p(z) = We will make use of this special representation of Hilbert polynomials and, in particular, of the sequence m 0,..., m r later. Their values will help to compute the set of generators of a special lexicographic ideal, which can be associated to a given Hilbert polynomial p(z). In the last part of this chapter we state a result to characterize Hilbert functions according to F.S. Macaulay. It will be useful in various proofs later within this thesis, for example when we will deal with special types of lexicographic ideals: By Macaulay s characterization it turns out that the growth of their Hilbert function is minimal. 1.3 Characterization of Hilbert functions As mentioned above, we will deal with the following question: Given a function h : Z Z, can one decide whether h : Z Z is the Hilbert function of R/I for I R a homogeneous ideal? As the heading suggests, we will give an answer to this question. The main result of this section, which answers the question stated above, was developed by F. S. Macaulay in 1927 (see [12] and [13]). Our presentation of the theory follows the treatment in the book Cohen-Macaulay rings by W. Bruns and J. Herzog (see [2], Chapter 4.2 ). ( ) i Note in the following that := 0 whenever i < j. j

25 1.3. CHARACTERIZATION OF HILBERT FUNCTIONS 15 Lemma Let d N. Every number a N can be uniquely written in the form ( ) ( ) ( ) kd kd 1 ks a = , d d 1 s where k d > k d 1 >... > k s s > 0 for s, k s,..., k d N. Proof. We have to prove existence and uniqueness of the representation. Existence. We proceed by induction on d. The case d = 1 is clear. Now let d > 1 and k d := max { k k N, ( ( ) ) } k kd d a. If we have a =, then take s := d and the rest ( ) d ( ) kd follows. Otherwise, we have a > and thus a kd := a > 0. By induction d d d 1 ( ) hypothesis, we obtain k d 1,..., k s N, k d 1 >... > k s > 0, such that a ki =. i i=s d ( ) ki Thus, we have a = and it remains to show that k d > k d 1. Since k d is maximal i ( ) i=s ( ) kd kd + 1 with a, we get > a and it follows that d d ( ) kd 1 a < d 1 ( ) kd + 1 d ( ) ( ) kd kd =, d d 1 and since the denominator of the binomial coefficients on the left and on the right are equal, it follows k d 1 < k d. Uniqueness. We prove by induction on a: If a can be written as ( ) ( ) ( ) kd kd 1 ks a = , d d 1 s then k d = max { k k N, ( } k d) a. In the case a = 1, we can only choose k d = d. Now, let a > 1. We proceed by induction on d. In the( case d ) = 1, we obtain k d = k 1 = a, and we are done. Let us assume that d > 1 kd + 1 and a. Then, we get d d 1 a > a := j=s ( ) kj = a j = ( kd ( kd d 1 ) ( ) kd + 1 d d ) ( ) kd 1 + 1, d 1 ( ) kd d

26 16 CHAPTER 1. NOTATIONS AND PREREQUISITES ( ) kd but > a by induction hypothesis, which is a contradiction. d 1 Since we have proved k d = max{k k N, ( k d) a }, one can determine kd, k d 1,..., k s one by one. Definition Let d, a N. We call a = ( ) ( ) kd kd d d 1 ( ) ks, s where k d > k d 1 >... > k s s > 0 for s, k s,..., k d N, the d-th Macaulay representation of a. We have a look at an example: ( ) ( ) 5 6 Example Let d := 3 and a := 12. Since = and = 20 > 12, we ( ) 3( ) ( 3) have to choose k 3 = 5. From 12 = 2 and = 1 2 < = 3, we obtain ( ) ( ) 3 2 ( 2) ( ) ( ) k 2 = 2. Finally 12 = 1 provides k 1 = 1. Thus, 12 = + + is the third Macaulay representation of 12. Definition For a, d N we define a d := ( ) kd d + 1 ( ) ( ) kd kd 1 where a = d d 1 case a = 0, we set 0 d := 0. ( ) kd d ( ) ks + 1, s + 1 ( ) ks is the d-th Macaulay representation of a. In the s Example ( ) ( ) ( In ) the terminology of the preceding Example 1.15, we obtain: 12 3 = = With this new terminology, we can characterize Hilbert functions, as mentioned above. Theorem (Macaulay s characterization of Hilbert functions) Let h : Z Z be a function. Then h = h R/I for a Hilbert function h R/I and some homogeneous ideal I R, if and only if, h(j) = 0 for all j < 0, h(0) = 1 and 0 h(j + 1) h(j) j for j > 0. The theorem will not be proved within this thesis. For a proof, see [2]. Instead, we will look at an example.

27 1.3. CHARACTERIZATION OF HILBERT FUNCTIONS 17 Example Let h : Z Z and h : Z Z be functions defined by 0 if j < 0 0 if j < 0 1 if j = 0 4 if j = 1 10 if j = 2 h(j) = 12 if j = 3 16 if j = 4 1 if j = 5 0 if j > 5 1 if j = 0 4 if j = 1 10 if j = 2 and h(j) = 12 if j = 3 18 if j = 4 1 if j = 5 0 if j > 5 Then h is the Hilbert function of R/I for some homogeneous ideal I, since all values fulfill the conditions of Theorem In contrast to this, h cannot be a Hilbert function of any such ring, since by Example 1.17 we have 12 3 = 17 and 18 = h(4) > h(3) 3 = 17, which contradicts the last condition on h of Theorem 1.18 (the value, which contradicts 1.18 is underlined in the definition of h above). Remark In 1.3 we defined lexicographic ideals. For a lexicographic ideal L R, which is generated by monomials of degree d for some d 1, we get h R/L (d + 1) = h R/L (d) d. This follows from [2], Chapter 4.2. A rather informal argument, which gives a hint, why the equation above is valid, is the following: We know, [L] d is generated by the r greatest monomials of [R] d for some r 1. We write the exponent vectors of these monomials in descending lexicographic order (d, 0, 0,..., 0, 0), (d 1, 1, 0,..., 0, 0), (d 1, 0, 1,..., 0, 0),..., (d 1, 0, 0,..., 1, 0), (d 1, 0, 0,..., 0, 1), (d 2, 2, 0,..., 0, 0), (d 2, 1, 1,..., 0, 0),..., (d 2, 1, 0,..., 1, 0), (d 2, 1, 0,..., 0, 1),..., v up to some vector v, the exponent vector of the smallest monomial of degree d in L. Now we consider the exponent vectors of all monomials of degree d + 1, which can be obtained from those above. This yields (d + 1, 0,..., 0, 0), (d, 1,..., 0, 0),..., (d, 0, 0,..., 1, 0), (d, 0, 0,..., 0, 1), (d, 1, 0,..., 0, 0), (d 1, 2, 0,..., 0, 0),..., (d 1, 1, 0,..., 1, 0), (d 1, 1, 0,..., 0, 1), (d, 0, 1,..., 0, 0), (d 1, 1, 1,..., 0, 0),..., (d 1, 0, 1,..., 1, 0), (d 1, 0, 1,..., 0, 1),. (d, 0, 0,..., 1, 0), (d 1, 1, 0,..., 1, 0),..., (d 1, 0, 1,..., 1, 0), (d 1, 0, 0,..., 1, 1), (d, 0, 0,..., 0, 1), (d 1, 1, 0,..., 0, 1),..., (d 1, 0, 1,..., 0, 1), (d 1, 0, 0,..., 0, 2),.

28 18 CHAPTER 1. NOTATIONS AND PREREQUISITES Those vectors, which have been underlined, are still contained in one of the rows above. In the second row, the first vector also appears in the row above, in the third row, the first vector also appears in the first row and the second vector also appears in the second. In the fourth row, the first vector appears in the first row, the second vector in the second and the third vector in the third row. Hence, by this combinatorial observation, we conclude that the number of monomials of degree d + 1, which can be obtained from a lexicographic set of r monomials of degree d, is the minimal number of monomials, which can be obtained from any set of r monomials of degree d in R. Consequently, any other set of r monomials of degree d will at least yield as many monomials of degree d + 1 as the set above. This indicates that the Hilbert function h L has minimal growth (from degree d up to degree d+1). Hence, the Hilbert function h R/L (j) must have maximal growth. Since the maximal value of h R/L (d+1) can be at most h R/L (d) d (by Theorem 1.18), we obtain h R/L (d+1) = h R/L (d) d. Example Consider the ideals L := (x 2 0, x 0 x 1, x 0 x 2 ) and I := (x 2 0, x 0 x 1, x 2 1) in R := K[x 0, x 1, x 2 ]. L is lexicographic, because it is generated by the three largest monomials of degree 2. Since x 0 x 2 > x 2 1 in lexicographic order and x 0 x 2 / I, I is not a lexicographic ideal. First, we determine the set of monomials in [L] 3 : Multiplication of x 2 0 by x 0, x 1, x 2 provides x 3 0, x 2 0x 1, x 2 0x 2. The monomial x 0 x 1 yields x 2 0x 1, x 0 x 2 1, x 0 x 1 x 2. Finally, we obtain from x 0 x 2 the three monomials x 2 0x 2, x 0 x 1 x 2 and x 0 x 2 2 (those, who appeared before, have been underlined as in the remark above). Hence, [L] 3 is generated by the six monomials x 3 0, x 2 0x 1, x 2 0x 2, x 0 x 2 1, x 0 x 1 x 2, x 0 x 2 2. In contrast to this, [I] 3 is generated by the seven monomials x 3 0, x 2 0x 1, x 2 0x 2, x 0 x 1 x 2, x 2 0x 1, x 3 1, x 2 1x 2.

29 Chapter 2 Stable ideals In this chapter, we introduce the class of stable ideals of the polynomial ring R := K[x 0,..., x n ]. We will state some results, which enable us to compute the saturation or even the Hilbert series and the Hilbert polynomial of a stable ideal in an easy way. Additionally, we consider special lexicographic ideals associated to a given Hilbert series or a given Hilbert polynomial. It turns out that these lexicographic ideals are also saturated and stable. The lexicographic ideals, which can be associated to a given Hilbert polynomial, will play an important role within the algorithm to determine all saturated stable ideals to a given Hilbert polynomial. As an appendix to this chapter, we present an application to the so-called Gotzmann s regularity Theorem (see [5], Chapter 3 ). 2.1 Borel-fixed ideals In order to define stable ideals we have to describe the action of special matrices on elements of the set M of all monomials of R. This first leads us to define Borel-fixed ideals. Definition 2.1. Let G := GL(n + 1, K) denote the multiplicative group of all invertible (n + 1) (n + 1)-matrices over K. For a matrix A = (a ij ) G and j {0,..., n}, we define the action of A on x j by A(x j ) := n a ij x i. i=0 Note that the rows and columns of A are indexed from 0 to n. For a monomial m = n R the action of A on m is defined to be the polynomial A(m), where j=0 x a j j A(m) := ( n n ) aj a ij x i. j=0 i=0 19

30 20 CHAPTER 2. STABLE IDEALS Example 2.2. Let R := K[x 0, x 1, x 2 ] and A = (a ij ) GL(2, K) with A = The action of A on the variables x 0, x 1, x 2 is given by A(x 0 ) = a 00 x 0 + a 10 x 1 + a 20 x 2 = x 0, A(x 1 ) = a 01 x 0 + a 11 x 1 + a 21 x 2 = 2x 0 + 2x 1, A(x 3 ) = a 02 x 0 + a 12 x 1 + a 22 x 2 = 3x 0 + 3x 1 + 3x 2. For the monomial m := x 0 x 2 1x 2 R the action of A on m is given by A(m) = x 0 (2x 0 + 2x 1 ) 2 (3x 0 + 3x 1 + 3x 2 ). In the following, we apply the matrix A to the monomial generators of an ideal. Let I := (x 3 0, x 2 0x 1, x 2 0x 2 ) R. Applying A to each of the monomial generators of I, we obtain the monomials A(x 3 0) = x 3 0, A(x 2 0x 1 ) = x 2 0 (2x 0 + 2x 1 ) = 2x x 2 0x 1, A(x 2 0x 2 ) = x 2 0 (3x 0 + 3x 1 + 3x 2 ) = 3x x 2 0x 1 + 3x 2 0x 2, and thereby the ideal I = (x 3 0, 2x x 2 0x 1, 3x x 2 0x 1 + 3x 2 0x 2 ). One can easily see that I I, since any monomial in the set of generators of I is a linear combination with coefficients in K of monomial generators of I. On the other hand we obtain x 3 0 I, x 2 0x 1 = 2x x 2 0x 1 2 x 3 0 I, x 2 0x 2 = 3x x 2 0x 1 + 3x 2 0x 2 x 2 3 0x 1 x 3 0 I. This shows I I and we conclude that the ideal I is fixed under the action of the matrix A on its monomial generators. If this holds for any invertible upper triangular (3 3) matrix, such an ideal will be called a stable ideal (in the case, when K is a field of characteristic zero). Definition 2.3. Let B denote the Borel-subgroup of G, i.e. the invertible upper triangular (n + 1) (n + 1)-matrices over K. A homogeneous ideal I R is called Borel-fixed if it is fixed under the action of matrices in B, in the sense that the application of any matrix of B on the generators of the ideal I reveals a set of polynomials generating the same ideal.

31 2.2. STABILITY OF BOREL-FIXED IDEALS Stability of Borel-fixed ideals In fact, the ideal I of the example above is Borel-fixed. Of course, there is no proof of this assertion in the example we do not give a proof using the definition above. Instead, we will present some further results, which allow us to find an easier characterization of Borel-fixed ideals. We will see: To decide, if an ideal I is Borel-fixed, it suffices to look at its generators. To present this result, we need some more theory on monomial orders on M. Remark and Definition 2.4. Let > be a monomial order on M. A weight function λ is a linear function R n+1 R. Every weight function induces a partial monomial order > λ on M, the weight order associated to λ, by setting for monomials m = x a 0 0 x a x an n, n = x b 0 0 x b x bn n M: m > λ n λ(a 0, a 1,..., a n ) > λ(b 0, b 1,..., b n ). Under a partial monomial order > λ we understand an order of monomials, satisfying the following two conditions for all monomials m 1, m 2, m 3 M: 1. If m 1 > λ m 2 and m 2 > λ m 3 then m 1 > λ m If m 1 > λ m 2 then m 1 m 3 > λ m 2 m 3. For a polynomial f R we denote by in λ f the sum of all terms of f, which are maximal with respect to > λ. Usually, we may think of λ being a function on M and write λ(m) for a monomial m = x a x an n M instead of λ(a 0,..., a n ). By setting λ i := λ(x i ), i = 0,..., n, we identify λ with the vector (λ 0,..., λ n ) and call this vector the associated weight vector to λ. Furthermore, we call we call λ(m) the weight of m. In the following, we will proceed in two steps: The first step is to show that each Borelfixed ideal is a monomial ideal (Note that the original definition of a Borel-fixed ideal does not contain this assumption). In the second step we characterize Borel-fixed ideals by a special combinatorial division property, which must be fulfilled by its generators. The following theorem is taken from [6], Theorem (a), p Theorem 2.5. Let I R be a homogeneous ideal. I is a monomial ideal if and only if I is fixed under the action of all diagonal matrices in G. Proof. = Let I R be a monomial ideal, and I g := {m 1,..., m s } M its generating

32 22 CHAPTER 2. STABLE IDEALS set, where m i = x a i x a in n. Let d 0,..., d n K\{0}, and D G the diagonal matrix d d D = d n d n Application of D to m j provides D(m j ) = n (d i x i ) a i = i=0 n i=0 d a i i m j. Consequently, the set {D(m 1 ),..., D(m s )} is given by {c 1 m 1,..., c s m s } where c 1,..., c s K\{0}. Both sets, I g and {c 1 m 1,..., c s m s }, generate the same ideal I. = Let I R be a homogeneous ideal, which is fixed under the action of all invertible (n + 1) (n + 1) diagonal matrices over K. Let f I. It suffices to show that every monomial of f is an element of I. One can prove that there is a weight function λ, such that in λ f is a single monomial of f, i.e. in λ f is the only monomial in f, which is maximal according to the monomial order > λ. The aim is to show in λ f I. Let w be the weight of in λ f and λ i := λ(x i ) for 0 i n. Application of the diagonal matrix D c G, c K\{0}, defined by c λ c λ D c = c λn on f will replace each variable x i in f by c λi x i, 0 i n. For w is the weight of in λ f, the application of D c to f provides a multiplication of in λ f with the factor c w. Since in λ f is the maximal monomial in f according to > λ, all other terms of f are multiplied by strictly less-negative powers of c. Therefore we write c w D c (f) = in λ f + c P (c, x) for a polynomial P in c and x. For every c 0 the matrix D c is invertible. Since f I and I is fixed under the action of D c this yields in λ f + c P (c, x) I for every c 0. For I is a Zariski closed subset, one may infer that in λ f + c P (c, x) I holds even for c = 0. Consequently, in λ f I and this proves the assertion. The preceding result can be proved in a more general setting. The field K in our context is of characteristic zero, but the theorem also holds for a field of characteristic p 0. The proof of this more general version is nearly the same as the proof presented above. It can be found in [6], proof of Theorem 15.23, pp. 356,357. Corollary 2.6. Every Borel-fixed ideal I R is a monomial ideal. Proof. If I is Borel-fixed, then I is fixed under all upper triangular matrices in G. In particular, I is fixed under all diagonal matrices in G. By the preceding theorem I must be monomial.

33 2.2. STABILITY OF BOREL-FIXED IDEALS 23 Now, we prove the central result of this section. It is again taken from [6], Theorem (b), p Theorem 2.7. (Stability of Borel-fixed ideals) Let I R be a homogeneous ideal. Then I is Borel-fixed if and only if the following condition holds for every monomial generator m of I: If m is divisible by x t j and no higher power of x j, then ( xi ) s m I for all i < j and s t. x j Proof. = Let I be Borel-fixed and let U c = (u c ij) G, c K, be an elementary upper triangular matrix, i.e. 1 if i = j u c ij = c if i = k and j = l, 0 in all other cases where 0 k l n. Now write m = x t l m for some m M not divisible by x l. The action of U c on m is given by U c (m) = (c x k + x l ) t m = t s=0 ( ) t s ( ) s c xk m. ( ) The first identity U c (m) = (c x k + x l ) t m is valid since all diagonal entries of the matrix U c are given by u c ii = 1, 0 i n. Consequently, the application of U c to a variable x i, i l, replaces x i by itself. The l-th column of U c contains another entry, the element c. So the variable x l is replaced by (c x k + x l ) for c is the entry of U c in the k-th row and l-th column. This shows U c (m) = (c x k + x l ) t m. The second part of the equation ( ) follows from t s=0 ( ) t s ( ) s c xk m = x l = s=0 s=0 x l t ( ) ( t c xk s x l t ( ) t s = (c x k + x l ) t m. ) s x t l m (c x k ) s x t s l m Since I is fixed under the action of U c, this provides U c (m) ( I. ) Since I is a homogeneous s c xk ideal, every term of U c (m) must be an element of I, i.e. m I. The case c = 1 proves the assertion. = Now let I be a monomial ideal fulfilling the division property of the theorem. We have to show that I is Borel-fixed. The formula ( ) above shows that for every generator x l

34 24 CHAPTER 2. STABLE IDEALS m M of I the term U c (m) is contained in I (if U c is an upper triangular matrix). Since I is a monomial ideal (i.e. I is homogeneous, especially), it follows I = U c (I). Since the set of all upper triangular matrices of the form U c and all diagonal matrices in G forms a basis of all upper triangular matrices of G, I is fixed under the action of all upper triangular matrices of G. This characterization of Borel-fixed ideals will be used in the following sections of this thesis and we will call an ideal, which fulfills the condition of the preceding Theorem 2.7, a stable ideal. Definition 2.8. Let K be a field of characteristic zero and let R := K[x 0,..., x n ]. A monomial ideal I R is called a stable ideal, if for every monomial m = x a 0 0 x a x an n of I we have for i < j and s a j. x a x a i 1 i 1 xa i+s i x a i+1 i+1... xa j 1 j 1 xa j s j x a j+1 j+1... xan n I Remark 2.9. Indeed, it is easy to see that the condition of Definition 2.8 is equivalent to the one in Theorem 2.7: Any stable ideal fulfills the condition of Theorem 2.7. On the other hand, let I R be a monomial ideal fulfilling the condition of Theorem 2.7. We have to show that I is stable. Let m I be a monomial. Then there is a monomial generator m I and a monomial m M, such that m = m m. Let x k, k > 0, be a variable dividing m. It suffices to show that x l m I for 0 l < k. If x k divides m, we have x k x l m x k = m hence, x l m x k = ) (x l m I, since m I. If x k divides m, we know x l m I and, x k ) x k (x l m m I, which proves the assertion. x k Example To come back to Example 2.2 of the preceding section, we can easily see that the ideal I = (x 3 0, x 2 0x 1, x 2 0x 2 ) is stable, since x 2 0x 1 x 1 x 0 = x 3 0 I, x 2 0x 2 x 2 x 1 = x 2 0x 1 I and x2 0x 2 x 2 x 0 = x 3 0 I. In the following, we will present an easy way to compute the saturation of a stable ideal. 2.3 Saturation of stable ideals In order to decide, whether a given stable ideal is saturated and, if this is not the case, to compute its saturation, we have to examine the associated prime ideals in the primary decomposition of the stable ideal. We will prove that all associated prime ideals of a given stable ideal are of the form (x 0,..., x i ) for an i 0. Within the proof of this assertion, we will make use of a theorem by Bayer and Stillman (see [6], Proposition ).

35 2.3. SATURATION OF STABLE IDEALS 25 Theorem (Bayer and Stillman) Let I R be a stable ideal. Then for every 0 j n, s 0. I : x s j = I : (x 0,..., x j ) s Proof. Since we have I : (x 0,..., x j ) s = {f f R, f g I for all g (x 0,..., x j ) s } and I : x s j = {f f R, f g I for all g (x j ) s }, the inclusion I : (x 0,..., x j ) s I : x s j is clear. Since I, (x 0,..., x j ) s and (x j ) s are monomial ideal, I : x s j and I : (x 0,..., x j ) s are also monomial ideals. Let m be a monomial in I : (x j ) s. Then x s j m I. Because I is stable, we obtain x a x a j j m I for all a 0,..., a j N 0 with shows I : x s j I : (x 0,..., x j ) s, and we are done. j a i = s by Theorem 2.7. This Lemma Let I R be a stable ideal. Then all associated prime ideals p R of I are generated by a sequence of variables of the form x 0,..., x j, j n. Proof. Since I is a monomial ideal, every associated prime ideal p of I is generated by a subset of {x 0,..., x n }. Let j be the maximal index such that x j p. We have to show that x i p for all 0 i j. Since p is an associated prime of I, there is an f R with p = I : f. Then x j p provides x j f I and f I : x j. For I is stable, we know from the preceding theorem that I : x j = I : (x 0,..., x j ). Thus, we have f I : (x 0,..., x j ), i.e. x i f I and therefore x i I : f = p for all 0 i j. The saturation of a stable ideal is computed due to the following result: Theorem (Saturation of stable ideals) Let I R be a stable ideal and let I g := {m 1,..., m s } M denote a set of minimal generators of I. The ideal sat xn (I) can be obtained from I by setting x n := 1 in every monomial of I g. Proof. If I is saturated, there is a primary decomposition I = q 1... q k with primary ideals q i m = (x 0,..., x n ) for 1 i s. The preceding lemma yields Rad q i = p i = (x 0,..., x li ), 1 i k, l i n. Since I is saturated, the variable x n cannot appear as a generator of any p i. Consequently, none of the elements of I g can contain the variable x n. On the other hand, if x n does not appear in any element of I g, x n cannot appear in any generator of q 1,..., q k (this follows mainly from the algorithm of computing the primary decomposition of monomial ideals, where the main aspect is to compute greatest common divisors and smallest common multiples of monomial generators). Since x n does not appear in any of the ideals q 1,..., q k, it cannot appear in p 1 = Rad q 1,..., p k = Rad q k. In the case that x k n I g is a generator of the ideal I, the saturation of the ideal I is the whole ring R. Thus, the ring R/ sat xn (I) describes the empty set (viewed as scheme), which is the trivial case and shall be of no further interest to us. i=0