SINGULAR Introduction to Commutative Algebra

Gert-Martin Greuel, Gerhard P ster A Singular Introdu tion to Commutative Algebra Mathemati s { Monograph (English) wit...

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Gert-Martin Greuel, Gerhard P ster

A Singular Introdu tion to Commutative Algebra Mathemati s { Monograph (English) with ontributions by Olaf Ba hmann, Christoph Lossen and Hans S h onemann

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Bar elona Budapest

To Ursula, Ursina, Jos ha, Bastian, Wanja, Gris ha G.{M. G.

To Marlis G. P.

Prefa e

In theory there is no dieren e between theory and pra ti e. In pra ti e there is. Yogi Berra

A SINGULAR Introdu tion to Commutative Algebra oers a rigorous intro-

du tion to ommutative algebra and, at the same time, provides algorithms and omputational pra ti e. In this book, we do not separate the theoreti al and the omputational part. Coin identally, as new on epts are introdu ed, it is onsequently shown, by means of on rete examples and general pro edures, how these on epts are handled by a omputer. We believe that this

ombination of theory and pra ti e will provide not only a fast way to enter a rather abstra t eld but also a better understanding of the theory, showing

on urrently how the theory an be applied. We exemplify the omputational part by using the omputer algebra system Singular, a system for polynomial omputations, whi h was developed in order to support mathemati al resear h in ommutative algebra, algebrai geometry and singularity theory. As the restri tion to a spe i system is ne essary for su h an exposition, the book should be useful also for users of other systems (su h as Ma aulay2 and CoCoA ) with similar goals. Indeed, on e the algorithms and the method of their appli ation in one system is known, it is usually not diÆ ult to transfer them to another system. The hoi e of the topi s in this book is largely motivated by what we believe is most useful for studying ommutative algebra with a view toward algebrai geometry and singularity theory. The development of ommutative algebra, although a mathemati al dis ipline in its own right, has been greatly in uen ed by problems in algebrai geometry and, onversely, ontributed signi antly to the solution of geometri problems. The relationship between both dis iplines an be hara terized by saying that algebra provides rigour while geometry provides intuition. In this onne tion, we pla e omputer algebra on top of rigour, but we should like to stress its limited value if it is used without intuition. During the past thirty years, in ommutative algebra, as in many parts of mathemati s, there has been a hange of interest from a most general theo-

II

Prefa e

reti al setting towards a more on rete and algorithmi understanding. One of the reasons for this was that new algorithms, together with the development of fast omputers, allowed non{trivial omputations, whi h had been intra table before. Another reason is the growing belief that algorithms an

ontribute to a better understanding of a problem. The human idea of \understanding", obviously, depends on the histori al, ultural and te hni al status of the so iety and, nowadays, understanding in mathemati s requires more and more algorithmi treatment and omputational mastering. We hope that this book will ontribute to a better understanding of ommutative algebra and its appli ations in this sense. The algorithms in this book are almost all based on Grobner bases or standard bases. The theory of Grobner bases is by far the most important tool for

omputations in ommutative algebra and algebrai geometry. Grobner bases were introdu ed originally by Bu hberger as a basis for algorithms to test the solvability of a system of polynomial omputations, to ount the number of solutions (with multipli ities) if this number is nite and, more algebrai ally, to ompute in the quotient ring modulo the given polynomials. Sin e then, Grobner bases have played an important role for any symboli omputations involving polynomial data, not only in mathemati s. We present, right at the beginning, the theory of Grobner bases and, more generally, standard bases, in a somewhat new avour.

Synopsis of the Contents of this Book From the beginning, our aim is to be able to ompute ee tively in a polynomial ring as well as in the lo alization of a polynomial ring at a maximal ideal. Geometri ally, this means that we want to ompute globally with (aÆne or proje tive) algebrai varieties and lo ally with its singularities. In other words, we develop the theory and tools to study the solutions of a system of polynomial equations, either globally or in a neighbourhood of a given point. The rst two hapters introdu e the basi theories of rings, ideals, modules and standard bases. They do not require more than a ourse in linear algebra, together with some training, to follow and do rigorous proofs. The main emphasis is on ideals and modules over polynomial rings. In the examples, we use a few fa ts from algebra, mainly from eld theory, and mainly to illustrate how to use Singular to ompute over these elds. In order to treat Grobner bases, we need, in addition to the ring stru ture, a total ordering on the set of monomials. We do not require, as is the ase in usual treatments of Grobner bases, that this ordering be a well{ordering. Indeed, non{well{orderings give rise to lo al rings, and are ne essary for a

omputational treatment of lo al ommutative algebra. Therefore, we introdu e, at an early stage, the general notion of lo alization. Having this, we introdu e the notion of a (weak) normal form in an axiomati way. The standard basis algorithm, as we present it, is the same for any monomial ordering,

Prefa e

III

only the normal form algorithm diers for well{orderings, alled global orderings in this book, and for non{global orderings, alled lo al, respe tively mixed, orderings. A standard basis of an ideal or a module is nothing but a spe ial set of generators (the leading monomials generate the leading ideal), whi h allows the omputation of many invariants of the ideal or module just from its leading monomials. We follow the tradition and all a standard basis for a global ordering a Grobner basis. The algorithm for omputing Grobner bases is Bu hberger's elebrated algorithm. It was modi ed by Mora to ompute standard bases for lo al orderings, and generalized by the authors to arbitrary (mixed) orderings. Mixed orderings are ne essary to generalize algorithms (whi h use an extra variable to be eliminated later) from polynomial rings to lo al rings. As the general standard basis algorithm already requires slightly more abstra tion than Bu hberger's original algorithm, we present it rst in the framework of ideals. The generalization to modules is then a matter of translation after the reader has be ome familiar with modules. Chapter 2 also

ontains some less elementary on epts su h as tensor produ ts, syzygies and resolutions. We use syzygies to give a proof of Bu hberger's riterion and, at the same time, the main step for a onstru tive proof of Hilbert's syzygy theorem for the (lo alization of the) polynomial ring. These rst two hapters nish with a olle tion of methods on how to use standard bases for various

omputations with ideals and modules, so{ alled \Grobner basi s". The next four hapters treat some more involved but entral on epts of

ommutative algebra. We follow the same method as in the rst two hapters, by onsequently showing how to use omputers to ompute more ompli ated algebrai stru tures as well. Naturally, the presentation is a little more ondensed, and the veri ation of several fa ts of a rather elementary nature are left to the reader as an exer ise. Chapter 3 treats integral losure, dimension theory and Noether normalization. Noether normalization is a ornerstone in the theory of aÆne algebras, theoreti ally as well as omputationally. It relates aÆne algebras, in a

ontrolled manner, to polynomial algebras. We apply the Noether normalization to develop the dimension theory for aÆne algebras, to prove the Hilbert Nullstellensatz and E. Noether's theorem that the normalization of an aÆne ring (that is, the integral losure in its total ring of fra tions) is a nite extension. For all this, we provide algorithms and on rete examples on how to

ompute them. A highlight of this hapter is the algorithm to ompute the non{normal lo us and the normalization of an aÆne ring. This algorithm is based on a riterion due to Grauert and Remmert, whi h had es aped the

omputer algebra ommunity for many years, and was redis overed by T. de Jong. The hapter ends with an extra se tion ontaining some of the larger pro edures, written in the Singular programming language. Chapter 4 is devoted to primary de omposition and related topi s su h as the equidimensional part and the radi al of an ideal. We start with the

IV

Prefa e

usual, short and elegant but not onstru tive proof, of primary de omposition of an ideal. Then we present the onstru tive approa h due to Gianni, Trager and Za harias. This algorithm returns the primary ideals and the asso iated primes of an ideal in the polynomial ring over a eld of hara teristi 0, but also works well if the hara teristi is suÆ iently large, depending on the given ideal. The algorithm, as implemented in Singular is often surprisingly fast. As in Chapter 3, we present the main pro edures in an extra se tion. In ontrast to the relatively simple existen e proof for primary de omposition, it is extremely diÆ ult to a tually de ompose even quite simple ideals, by hand. The reason be omes lear when we onsider the onstru tive proofs whi h are all quite involved, and whi h use many non{obvious results from ommutative algebra, eld theory and Grobner bases. Indeed, primary de omposition is an important example, where we learn mu h more from the

onstru tive proof than from the abstra t one. In Chapter 5 we introdu e the Hilbert fun tion and the Hilbert polynomial of graded modules together with its appli ation to dimension theory. The Hilbert polynomial, respe tively its lo al ounterpart, the Hilbert{ Samuel polynomial, ontains important information about a homogeneous ideal in a polynomial ring, respe tively an arbitrary ideal, in a lo al ring. The most important one, besides the dimension, is the degree in the homogeneous ase, respe tively the multipli ity in the lo al ase. We prove that the Hilbert ({Samuel) polynomial of an ideal and of its leading ideal oin ide, with respe t to a degree ordering, whi h is the basis for the omputation of these fun tions. The hapter nishes with a proof of the Ja obian riterion for aÆne K {algebras and its appli ation to the omputation of the singular lo us, whi h uses the equidimensional de omposition of the previous hapter; other algorithms, not using any de omposition, are given in the exer ises to Chapter 7. Standard bases were, independent of Bu hberger, introdu ed by Hironaka in onne tion with resolution of singularities and by Grauert in onne tion with deformation of singularities, both for ideals in power series rings. We introdu e ompletions and formal power series in Chapter 6. We prove the

lassi al Weierstra preparation and division theorems and Grauert's generalization of the division theorem to ideals, in formal power series rings. Besides this, the main result here is that standard bases of ideals in power series rings an be omputed if the ideal is generated by polynomials. This is the basis for omputations in lo al analyti geometry and singularity theory. The last hapter, Chapter 7, gives a short introdu tion to homologi al algebra. The main purpose is to study various aspe ts of depth and atness. Both notions play an important role in modern ommutative algebra and algebrai geometry. Indeed, atness is the algebrai reason for what the an ient geometers alled \prin iple of onservation of numbers\, as it guarantees that

ertain invariants behave ontinuously in families of modules, respe tively varieties. After studying and showing how to ompute Tor{modules, we use Fit-

Prefa e

V

ting ideals to show that the at lo us of a nitely presented module is open. Moreover, we present an algorithm to ompute the non{ at lo us and, even further, a attening strati ation of a nitely presented module. We study, in some detail, the relation between atness and standard bases, whi h is somewhat subtle for mixed monomial orderings. In parti ular, we use atness to show that, for any monomial ordering, the ideal and the leading ideal have the same dimension. In the nal se tions of this hapter we use the Koszul omplex to study the relation between the depth and the proje tive dimension of a module. In parti ular, we prove the Auslander{Bu hsbaum formula and Serre's hara terization of regular lo al rings. These an be used to ee tively test the Cohen{Ma aulay property and the regularity of a lo al K {algebra. The book ends with two appendi es, one on the geometri ba kground and the se ond one on an overview on the main fun tionality of the system Singular. The geometri ba kground introdu es the geometri language, to illustrate some of the algebrai onstru tions introdu ed in the previous hapters. One of the obje ts is to explain, in the aÆne as well as in the proje tive setting, the geometri meaning of elimination as a method to ompute the ( losure of the) image of a morphism. Moreover, we explain the geometri meaning of the degree and the multipli ity de ned in the hapter on the Hilbert Polynomial (Chapter 5), and prove some of its geometri properties. This appendix ends with a view towards singularity theory, just tou hing on Milnor and Tjurina numbers, Arnold's lassi ation of singularities, and deformation theory. All this, together with other on epts of singularity theory, su h as Puiseux series of plane urve singularities and monodromy of isolated hypersurfa e singularities, and many more, whi h are not treated in this book, an be found in the a

ompanying libraries of Singular. The se ond appendix gives a ondensed overview of the programming language of Singular, data types, fun tions and ontrol stru ture of the system, as well as of the pro edures appearing in the libraries distributed with the system. Moreover, we show by three examples (Maple, Mathemati a, MuPAD), how Singular an ommuni ate with other systems.

How to Use the Text The present book is based on a series of le tures held by the authors over the past ten years. We tried several ombinations in ourses of two, respe tively four, hours per week in a semester (12 { 14 weeks). There are at least four aspe ts on how to use the text for a le ture: (A) Fo us on omputational aspe ts of standard bases, and syzygies. A possible sele tion for a two{hour le ture is to treat Chapters 1 and 2

ompletely (possibly omitting 2.6, 2.7). In a four{hour ourse one an treat, additionally, 3.1 { 3.5 together with either 4.1 { 4.3 or 4.1 and 5.1 { 5.3.

VI

Prefa e

(B) Fo us on appli ations of methods based on standard basis, respe tively syzygies, for treating more advan ed problems su h as primary de omposition, Hilbert fun tions, or atness (regarding the standard basis, respe tively syzygy, omputations as \bla k boxes"). In this ontext a two{hour le ture ould over Se tions 1.1 { 1.4 (only treating global orderings), 1.6 (omitting the algorithms), 1.8, 2.1, Chapter 3 and Se tion 4.1. A four{hour le ture ould treat, in addition, the ase of lo al orderings, Se tion 1.5, and sele ted parts of Chapters 5 and 7. (C) Fo us on the theory of ommutative algebra, using for examples and experiments.

Singular

as a tool

Here a two{hour ourse ould be based on Se tions 1.1, 1.3, 1.4, 2.1, 2.2, 2.4, 2.7, 3.1 { 3.5 and 4.1. For a four{hour le ture one ould hoose, additionally, Chapter 5 and Se tions 7.1 { 7.4. (D) Fo us on geometri aspe ts, using

Singular

as a tool for examples.

In this ontext a two{hour le ture ould be based on Appendix A.1, A.2 and A.4, together with the needed on epts and statements of Chapters 1 and 3. For a four{hour le ture one is free to hoose additional parts of the appendix (again together with the ne essary ba kground from Chapters 1 { 7). Of ourse, the book may also serve as a basis for seminars and, last but not least, as a referen e book for omputational ommutative algebra and algebrai geometry.

Working with SINGULAR The original motivation for the authors to develop a omputer algebra system in the mid eighties, was the need to ompute invariants of ideals and modules in lo al rings, su h as Milnor numbers, Tjurina numbers, and dimensions of modules of dierentials. The question was whether the exa tness of the Poin are omplex of a omplete interse tion urve singularity is equivalent to the urve being quasihomogeneous. This question was answered by an early version of Singular: it is not [149℄. In the sequel, the development of Singular was always in uen ed by mathemati al problems, for instan e, the famous Zariski onje ture, saying that the onstan y of the Milnor number in a family implies onstant multipli ity [90℄. This onje ture is still unsolved. En losed in the book one nds a CD with folders EXAMPLES, LIBRARIES, MAC, MANUAL, UNIX and WINDOWS. The folder EXAMPLES ontains all Singular Examples of the book, the pro edures and the links to Mathemati a, Maple and MuPAD. The other folders ontain the Singular binaries for the respe tive platforms, the manual, a tutorial and the Singular libraries. Singular

an be installed following the instru tions in the INSTALL .html (or INSTALL .txt) le of the respe tive folder. We also should like to refer to the Singular homepage

Prefa e

VII

http://www.singular.uni-kl.de

whi h always oers the possibility to download the newest version of Singular, provides support for Singular users and a dis ussion forum. Moreover, one nds there a lot of useful information around Singular, for instan e, more advan ed examples and appli ations than provided in this book.

Comments and Corre tions We should like to en ourage omments, suggestions and orre tions to the book. Please send them to either of us: Gert{Martin Greuel greuelmathematik.uni-kl.de Gerhard P ster p stermathematik.uni-kl.de We also en ourage the readers to he k the web site for A SINGULAR In-

trodu tion to Commutative Algebra,

http://www.singular.uni-kl.de/Singular-book.html

This site will ontain lists of orre tions, respe tively of solutions for sele ted exer ises.

A knowledgements As is ustomary for textbooks, we use and reprodu e results from ommutative algebra, usually without any spe i attribution and referen e. However, we should like to mention that we have learned ommutative algebra mainly from the books of Zariski{Samuel [188℄, Nagata [146℄, Atiyah{Ma donald [4℄, Matsumura [125℄ and from Eisenbud's re ent book [51℄. The geometri ba kground and motivation, present at all times while writing this book, were laid by our tea hers Egbert Brieskorn and Herbert Kurke. The reader will easily re ognize that our book owes a lot to the admirable work of the above{mentioned mathemati ians, whi h we gratefully a knowledge. There remains only the pleasant duty of thanking the many people who have ontributed in one way or another to the preparation of this work. First of all, we should like to mention Christoph Lossen, who not only substantially improved the presentation but also ontributed to the theory as well as to proofs, examples and exer ises. The book ould not have been written without the system Singular, whi h has been developed over a period of about fteen years by Hans S honemann and the authors, with onsiderable ontributions by Olaf Ba hmann. We feel that it is just fair to mention these two as o{authors of the book, a knowledging, in this way, their ontribution as the prin ipal reators of the Singular system.1 1 \Software is hard. It's harder than anything else I've ever had to do." (Donald E. Knuth)

VIII

Prefa e

Further main ontributors to Singular in lude: W. De ker, A. FruhbisKruger, H. Grassmann, T. Keilen, K. Kruger, V. Levandovskyy, C. Lossen, M. Messollen, W. Neumann, W. Pohl, J. S hmidt, M. S hulze, T. Siebert, R. Stobbe, M. Wenk, E. Westenberger and T. Wi hmann, together with many authors of Singular libraries mentioned in the headers of the orresponding library. Proofreading was done by many of the above ontributors and, moreover, by Y. Drozd, T. de Jong, D. Popes u, and our students M. Bri kenstein, K. Dehmann, M. Kunte, H. Markwig and M. Olbermann. Last but not least, Pauline Bits h did the LATEX{typesetting of many versions of our manus ript and most of the pi tures were prepared by Thomas Keilen. We wish to express our heartfelt2 thanks to all these ontributors.

The book is dedi ated to our families, espe ially to our wives Ursula and Marlis, whose en ouragement and onstant support have been invaluable. Kaiserslautern, Mar h, 2002

Gert{Martin Greuel Gerhard P ster

2 The heart is displayed by using the program surf, see Singular Example A.1.1.

Contents

Prefa e : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

I

1. Rings, Ideals and Standard Bases : : : : : : : : : : : : : : : : : : : : : : : : :

1 1 9 19 30 38 44 53 67 67 69 71 74 75 77 79 80 81 84 86

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Rings, Polynomials and Ring Maps . . . . . . . . . . . . . . . . . . . . . . . Monomial Orderings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ideals and Quotient Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lo al Rings and Lo alization . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rings Asso iated to Monomial Orderings . . . . . . . . . . . . . . . . . . Normal Forms and Standard Bases . . . . . . . . . . . . . . . . . . . . . . . The Standard Basis Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . Operations on Ideals and their Computation . . . . . . . . . . . . . . . 1.8.1 Ideal Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Interse tion with Subrings . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.3 Zariski Closure of the Image . . . . . . . . . . . . . . . . . . . . . . . 1.8.4 Solvability of Polynomial Equations . . . . . . . . . . . . . . . . . 1.8.5 Solving Polynomial Equations . . . . . . . . . . . . . . . . . . . . . . 1.8.6 Radi al Membership . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.7 Interse tion of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.8 Quotient of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.9 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.10 Kernel of a Ring Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.11 Algebrai Dependen e and Subalgebra Membership . . .

2. Modules : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 91 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

Modules, Submodules and Homomorphisms . . . . . . . . . . . . . . . . Graded Rings and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard Bases for Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exa t Sequen es and free Resolutions . . . . . . . . . . . . . . . . . . . . . Computing Resolutions and the Syzygy Theorem . . . . . . . . . . . Modules over Prin ipal Ideal Domains . . . . . . . . . . . . . . . . . . . . . Tensor Produ t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operations on Modules and their Computation . . . . . . . . . . . . . 2.8.1 Module Membership Problem . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Interse tion with Free Submodules . . . . . . . . . . . . . . . . . .

91 114 118 128 139 153 167 177 177 179

X

Contents

2.8.3 2.8.4 2.8.5 2.8.6 2.8.7 2.8.8

Interse tion of Submodules . . . . . . . . . . . . . . . . . . . . . . . . Quotients of Submodules . . . . . . . . . . . . . . . . . . . . . . . . . . Radi al and Zerodivisors of Modules . . . . . . . . . . . . . . . . Annihilator and Support . . . . . . . . . . . . . . . . . . . . . . . . . . Kernel of a Module Homomorphism . . . . . . . . . . . . . . . . Solving Systems of Linear Equations . . . . . . . . . . . . . . . .

180 181 183 185 186 187

3. Noether Normalization and Appli ations : : : : : : : : : : : : : : : : : 193 3.1 3.2 3.3 3.4 3.5 3.6 3.7

Finite and Integral Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . The Integral Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Noether Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appli ations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Algorithm to Compute the Normalization . . . . . . . . . . . . . . Pro edures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

193 201 207 213 217 226 233

4. Primary De omposition and Related Topi s : : : : : : : : : : : : : : 241 4.1 4.2 4.3 4.4 4.5 4.6

The Theory of Primary De omposition . . . . . . . . . . . . . . . . . . . . Zero{dimensional Primary De omposition . . . . . . . . . . . . . . . . . Higher Dimensional Primary De omposition . . . . . . . . . . . . . . . The Equidimensional Part of an Ideal . . . . . . . . . . . . . . . . . . . . . The Radi al . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pro edures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

241 246 255 260 263 267

5. Hilbert Fun tion and Dimension : : : : : : : : : : : : : : : : : : : : : : : : : : 277 5.1 5.2 5.3 5.4 5.5 5.6 5.7

The Hilbert Fun tion and the Hilbert Polynomial . . . . . . . . . . . Computation of the Hilbert{Poin are Series . . . . . . . . . . . . . . . . Properties of the Hilbert Polynomial . . . . . . . . . . . . . . . . . . . . . . Filtrations and the Lemma of Artin{Rees . . . . . . . . . . . . . . . . . . The Hilbert{Samuel Fun tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chara terization of the Dimension of Lo al Rings . . . . . . . . . . . Singular Lo us . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

277 281 285 292 294 301 307

6. Complete Lo al Rings : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 315 6.1 6.2 6.3 6.4

Formal Power Series Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Weierstra Preparation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Standard bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

315 319 327 333

7. Homologi al Algebra : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 337 7.1 7.2 7.3 7.4 7.5

Tor and Exa tness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fitting Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lo al Criteria for Flatness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flatness and Standard Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

337 343 348 359 364

Contents

7.6 7.7 7.8 7.9

Koszul Complex and Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cohen{Ma aulay Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Further Chara terization of Cohen{Ma aulayness . . . . . . . . . . . Homologi al Chara terization of Regular Rings . . . . . . . . . . . . .

XI

371 384 390 399

Appendix : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 403 A. Geometri Ba kground : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 403 A.1 A.2 A.3 A.4 A.5 A.6 A.7 A.8 A.9

Introdu tion by Pi tures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . AÆne algebrai varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spe trum and AÆne S hemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proje tive Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proje tive S hemes and Varieties . . . . . . . . . . . . . . . . . . . . . . . . . Morphisms between Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proje tive Morphisms and Elimination . . . . . . . . . . . . . . . . . . . . Lo al versus Global Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

403 412 423 431 443 448 456 470 484

B. SINGULAR | A Short Introdu tion : : : : : : : : : : : : : : : : : : : : : 497 B.1 Downloading Instru tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Pro edures and Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4 Data Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.5 Fun tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.6 Control Stru tures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.7 System variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.8 Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.9 Singular and Maple . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.10 Singular and Mathemati a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.11 Singular and MuPAD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

497 500 504 506 512 528 529 530 546 550 551

Referen es : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 555 Glossary : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 565 Index : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 569

Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 Singular{Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589

XII

Contents

1. Rings, Ideals and Standard Bases

1.1 Rings, Polynomials and Ring Maps The on ept of a ring is probably the most basi one in ommutative and non{ ommutative algebra. Best known are the ring of integers Z and the polynomial ring K [x℄ in one variable x over a eld K . We shall now introdu e the general on ept of a ring with spe ial emphasis on polynomial rings.

De nition 1.1.1. (1) A ring is a set A together with an addition + : A A ! A, (a; b) 7! a + b, and a multipli ation : A A ! A, (a; b) 7! a b = ab, satisfying a) A, together with the addition, is an abelian group; the neutral element being denoted by 0 and the inverse of a 2 A by a; b) the multipli ation on A is asso iative, that is, (ab) = a(b ) and the distributive law holds, that is, a(b + ) = ab + a and (b + )a = ba + a, for all a; b; 2 A. (2) A is alled ommutative if ab = ba for a; b 2 A and has an identity if there exists an element in A, denoted by 1, su h that 1 a = a 1 for all a 2 A. In this book a ring always means a ommutative ring with identity. Be ause of (1) a ring annot be empty but it may onsist only of one element 0, this being the ase if and only if 1 = 0.

De nition 1.1.2. (1) A subset of a ring A is alled a subring if it ontains 1 and is losed under the ring operations indu ed from A. (2) u 2 A is alled a unit if there exists a u0 2 A su h that uu0 = 1. The set of units is denoted by A ; it is a group under multipli ation. (3) A ring is a eld if 1 = 6 0 and any non{zero element is a unit, that is A = A f0g. (4) Let A be a ring, a 2 A, then hai := faf j f 2 Ag.

Any eld is a ring, su h as Q (the rational numbers), or R (the real numbers), or C (the omplex numbers), or F p = Z=pZ (the nite eld with p elements

2


where p is a prime number, f. Exer ise 1.1.3) but Z (the integers) is a ring whi h is not a eld. Z is a subring of Q , we have Z = f1g, Q = Q r f0g. N Z denotes the set of nonnegative integers.

De nition 1.1.3. Let A be a ring. (1) A monomial in n variables (or indeterminates) x1 ; : : : ; xn is a power produ t

x = x11 : : : xnn ;

= (1 ; : : : ; n ) 2 N n :

The set of monomials in n variables is denoted by Mon(x1 ; : : : ; xn ) = Monn := fx j 2 N n g : Note that Mon(x1 ; : : : ; xn ) is a semigroup under multipli ation, with neutral element 1 = x01 : : : x0n . We write x j x if x divides x , whi h means that i i for all i and, hen e, x = x x for = 2 N n . (2) A term is a monomial times a oeÆ ient (an element of A),

ax = ax1 1 : : : xnn ; a 2 A : (3) A polynomial over A is a nite A{linear ombination of monomials, that is, a nite sum of terms,

f=

X

a x =

nite X

2

Nn

a1 :::n x1 1 : : : xnn ;

with a 2 A. For 2 N n , let jj := 1 + + n . The integer deg(f ) := maxfjj j a 6= 0g is alled the degree of f if f 6= 0; we set deg(f ) = 1 for f the zero polynomial. (4) The polynomial ring A[x℄ = A[x1 ; : : : ; xn ℄ in n variables over A is the set of all polynomials together with the usual addition and multipli ation: X

X

a x +

X

b x :=

1 ! 0 X a x b x A

:=

X

(a + b )x ;

X

0

X

+ =

1

a b A x :

A[x1 ; : : : ; xn ℄ is a ommutative ring with identity 1 = x01 : : : x0n whi h we identify with the identity element 1 2 A. Elements of A A[x℄ are alled

onstant polynomials , they are hara terized by having degree 0. A is alled the ground ring of A[x℄, respe tively the ground eld , if A is a eld.

1.1 Rings, Polynomials and Ring Maps

3

Note that any monomial is a term (with oeÆ ient 1) but, for example, 0 is a term but not a monomial. For us the most important ase is the polynomial ring K [x℄ = K [x1 ; : : : ; xn ℄ over a eld K . By Exer ise 1.3.1 only the non{zero

onstants are units of K [x℄, that is, K [x℄ = K = K r f0g. If K is an in nite eld, we an identify polynomials f 2 K [x1 ; : : : ; xn ℄ with their asso iated polynomial fun tion

f~ : K n

! K;

(p1 ; : : : ; pn ) 7

! f (p1 ; : : : ; pn ) ;

but for nite elds f~ may be zero for a non{zero f ( f. Exer ise 1.1.4). Any polynomial in n 1 variables an be onsidered as a polynomial in n variables (where the n{th variable does not appear) with the usual ring operations on polynomials in n variables. Hen e, A[x1 ; : : : ; xn 1 ℄ A[x1 ; : : : ; xn ℄ is a subring and it follows dire tly from the de nition of polynomials that

A[x1 ; : : : ; xn ℄ = (A[x1 ; : : : ; xn 1 ℄)[xn ℄ : Hen e, we an write f

2 A[x1 ; : : : ; xn ℄ in a unique way, either as f=

nite X

2

Nn

a x ; a 2 A

or as

f=

nite X

2N

f xn ; f

2 A[x1 ; : : : ; xn

1℄ :

The rst representation of f is alled distributive while the se ond is alled

re ursive .

Remark 1.1.4. Both representations play an important role in omputer al-

gebra. The pra ti al performan e of an implemented algorithm may depend drasti ally on the internal representation of polynomials (in the omputer). Usually the distributive representation is hosen for algorithms related to Grobner basis omputations while the re ursive representation is preferred for algorithms related to fa torization of polynomials.

De nition 1.1.5. A morphism or homomorphism of rings is a map ' : A ! B satisfying '(a + a0 ) = '(a) + '(a0 ), '(aa0 ) = '(a)'(a0 ), for all a; a0 2 A, and '(1) = 1. We all a morphism of rings also a ring map , and B is alled an A{algebra .1 We have '(a) = '(a 1) = '(a) 1. If ' is xed, we also write a b instead of '(a) b for a 2 A and b 2 B . 1 See also Example 2.1.2 and De nition 2.1.3.

4


Lemma 1.1.6. Let A[x1 ; : : : xn ℄ be a polynomial ring, : A ! B a ring map, C a B{algebra, and f1 ; : : : ; fn 2 C. Then there exists a unique ring map ' : A[x1 ; : : : ; xn ℄

!C

satisfying '(xi ) = fi for i = 1; : : : ; n and '(a) = (a) 1 2 C for a 2 A. P

Proof. Given any f = a x 2 A[x℄, then a ring map ' with '(xi ) = fi , and '(a) = (a) for a 2 A must satisfy (by De nition 1.1.5) '(f ) =

X

(a )'(x1 )1 : : : '(xn )n :

Hen e, ' is uniquely determined. Moreover, de ning '(f ) for f 2 A[x℄ by the above formula, it is easy to see that ' be omes a homomorphism, whi h proves existen e. We shall apply this lemma mainly to the ase where C is the polynomial ring B [y1 ; : : : ; ym ℄. In Singular one an de ne polynomial rings over the following elds: (1) (2) (3) (4) (5) (6) (7) (8)

the eld of rational numbers Q , nite elds Fp , p a prime number 32003, nite elds GF(pn ) with pn elements, p a prime, pn 215 , trans endental extensions of Q or F p , simple algebrai extensions of Q or F p , simple pre ision real oating point numbers, arbitrary pres ribed real oating point numbers, arbitrary pres ribed omplex oating point numbers.

For the de nitions of rings over elds of type (3) and (5) we use the fa t that for a polynomial ring K [x℄ in one variable x over a eld and f 2 K [x℄ r f0g the quotient ring K [x℄=hf i is a eld if and only if f is irredu ible , that is, f

annot be written as a produ t of two polynomials of lower degree ( f. Exer ise 1.1.5). If f is irredu ible and moni , then it is alled the minimal polynomial of the eld extension K K [x℄=hf i ( f. Example 1.1.8).

Remark 1.1.7. Indeed, the omputation over the above elds (1) { (5) is exa t, only limited by the internal memory of the omputer. Stri tly speaking,

oating point numbers, as in (6) { (8), do not represent the eld of real (or

omplex) numbers. Be ause of rounding errors, the produ t of two non{zero elements or the dieren e between two unequal elements may be zero (the latter ase is the more serious one sin e the individual elements may be very big). Of ourse, in many ases one an trust the result, but we should like to emphasize that this remains the responsibility of the user, even if one

omputes with very high pre ision.


5

Singular, eld elements have the type number but noti e that one an de ne and use numbers only in a polynomial ring with at least one variable

In

and a spe i ed monomial ordering. For example, if one wishes to ompute with arbitrarily big integers or with exa t arithmeti in Q , this an be done as follows:

SINGULAR Example 1.1.8 ( omputation in elds).

In the examples below we have used the degree reverse lexi ographi al ordering dp but we ould have used any other monomial ordering ( f. Se tion 1.2). A tually, this makes no dieren e as long as we do simple manipulations with polynomials. However, more ompli ated operations on ideals su h as the std or groebner ommand return results whi h depend very mu h on the hosen ordering. (1) Computation in the eld of rational numbers : ring A = 0,x,dp; number n = 12345/6789; n^5; // ommon divisors are an elled //-> 1179910858126071875/59350279669807543

Note: Typing just

123456789^5; will result in integer over ow sin e 123456789 is onsidered as an integer (ma hine integer of limited size)

and not as an element in the eld of rational numbers; however, also

orre t would be number(123456789)^5;. (2) Computation in nite elds : ring A1 = 32003,x,dp; number(123456789)^5; //-> 8705

//finite field Z/32003

ring A2 = (2^3,a),x,dp; //finite (Galois) field GF(8) //with 8 elements number n = a+a2; //a is a generator of the group //GF(8)-{0} n^5; //-> a6 //minimal polynomial of GF(8) minpoly; //-> 1*a^3+1*a^1+1*a^0 ring A3 = (2,a),x,dp; minpoly = a20+a3+1;

number n = a+a2;

//infinite field Z/2(a) of // hara teristi 2 //define a minimal polynomial //a^20+a^3+1 //now the ground field is //GF(2^20)=Z/2[a℄/, //a finite field

6

1. Rings, Ideals and Standard Bases n^5; //-> (a10+a9+a6+a5)

//with 2^20 elements //a is a generator of the group //GF(2^20)-{0}

Note: For omputation in nite elds Z=pZ, p 32003, respe tively

GF (pn ), pn 215 , one should use rings as A1 respe tively A2 sin e for these elds Singular uses look{up tables, whi h is quite fast. For other nite elds a minimal polynomial as in A3 must be spe i ed. A good

hoi e are the Conway polynomials ( f. [102℄). Singular does not, however, he k the irredu ibility of the hosen minimal polynomial. This an be done as in the following example. ring tst = 2,a,dp; fa torize(a20+a2+1,1); //-> _[1℄=a3+a+1 //not irredu ible! We have two fa tors //-> _[2℄=a7+a5+a4+a3+1 fa torize(a20+a3+1,1); //irredu ible //-> _[1℄=a20+a3+1

To obtain the multipli ities of the fa tors, use fa torize(a20+a2+1);. (3) Computation with real and omplex oating point numbers , 30 digits pre ision: ring R1 = (real,30),x,dp; number n = 123456789.0; n^5; // ompute with a pre ision of 30 digits //-> 0.286797186029971810723376143809e+41

Note: n5 is a number whose integral part has 41 digits (indi ated by e+41). However, only 30 digits are omputed.

ring R2 = ( omplex,30,I),x,dp;//I denotes imaginary unit number n = 123456789.0+0.0001*I; n^5; // omplex number with 30 digits pre ision //-> (0.286797186029971810723374262133e+41 +I*116152861399129622075046746710)

(4) Computation with rational numbers and parameters , that is, in Q (a; b; ), the quotient eld of Q [a; b; ℄: ring R3 = (0,a,b, ),x,dp; number n = 12345a+12345/(78b ); n^2; //->(103021740900a2b2 2+2641583100ab +16933225)/(676b2 2) n/9 ; //-> (320970ab +4115)/(234b 2)


7

We shall now show how to de ne the polynomial ring in n variables x1 ; : : : ; xn over the above mentioned elds K . We an do this for any n, but we have to spe ify an integer n rst. The same remark applies if we work with trans endental extensions of degree m; we usually all the elements t1 ; : : : ; tm of a trans endental basis (free) parameters . If g is any non{zero polynomial in the parameters t1 ; : : : ; tm , then g and 1=g are numbers in the orresponding ring. For further examples see the Singular Manual [93℄.

SINGULAR Example 1.1.9 ( omputation in polynomial rings). Let us reate polynomial rings over dierent elds. By typing the name of the ring we obtain all relevant information about the ring. ring A = 0,(x,y,z),dp; poly f = x3+y2+z2; //same as x^3+y^2+z^2 f*f-f; //-> x6+2x3y2+2x3z2+y4+2y2z2+z4-x3-y2-z2 Singular understands short (e.g., 2x2+y3) and long (e.g., 2*x^2+y^3) input.

By default the short output is displayed in rings without parameters and with one{letter variables, whilst the long output is used, for example, for indexed variables. The ommand short=0; for es all output to be displayed in the long format. Computations in polynomial rings over other elds follow the same pattern. Try ring R=32003,x(1..3),dp; ( nite ground eld), respe tively ring R=(0,a,b, ),(x,y,z,w),dp; (ground eld with parameters), and type R; to obtain information about the ring. The ommand setring allows swit hing from one ring to another, for example, setring A4; makes A4 the basering. We use Lemma 1.1.6 to de ne ring maps in Singular. Indeed, one has three possibilities, fet h, imap and map, to de ne ring maps by giving the name of the preimage ring and a list of polynomials f1 ; : : : ; fn (as many as there are variables in the preimage ring) in the urrent basering. The ommands fet h, respe tively imap, map an obje t dire tly from the preimage ring to the basering whereas fet h maps the rst variable to the rst, the se ond to the se ond and so on (hen e, is onvenient for renaming the variables), while imap maps a variable to the variable with the same name (or to 0 if it does not exist), hen e is onvenient for in lusion of sub{rings or for hanging the monomial ordering.

Note: All maps go from a prede ned ring to the basering.

8


SINGULAR Example 1.1.10 (methods for reating ring maps). map: preimage ring

! basering

(1) General de nition of a map : ring A = 0,(a,b, ),dp; poly f = a+b+ab+ 3; ring B = 0,(x,y,z),dp; map F = A, x+y,x-y,z;//map F from ring A (to basering B) //sending a -> x+y, b -> x-y, -> z poly g = F(f); //apply F g; //-> z3+x2-y2+2x

(2) Spe ial maps (imap, fet h): ring A1 = 0,(x,y, ,b,a,z),dp; imap(A,f); //imap preserves names of variables //-> 3+ba+b+a fet h(A,f); //fet h preserves order of variables //-> 3+xy+x+y

Exer ises 1.1.1. The set of units A of a ring A is a group under multipli ation. 1.1.2. The dire t sum of rings A B , together with omponent{wise addi-

tion and multipli ation is again a ring.

1.1.3. Prove that, for n 2 Z, the following are equivalent:

(1) Z=hni is a eld. (2) Z=hni is an integral domain. (3) n is a prime number. 1.1.4. Let K be a eld and f 2 K [x1 ; : : : ; xn ℄. Then f determines a polynomial fun tion fe : K n ! K , (p1 ; : : : ; pn ) 7! f (p1 ; : : : ; pn). (1) If K is in nite then f is uniquely determined by f~. (2) Show by an example that this is not ne essarily true for K nite. (3) Let K be a nite eld with q elements. Show that ea h polynomial f 2 K [x1 ; : : : ; xn ℄ of degree at most q 1 in ea h variable is already determined by the polynomial fun tion fe : K n ! K .

1.1.5. Let f 2 K [x℄ be a non{ onstant polynomial in one variable over the

eld K . f is alled irredu ible if f 62 K and if it is not the produ t of two polynomials of stri tly smaller degree. Prove that the following are equivalent:

1.2 Monomial Orderings

9

(1) K [x℄=hf i is a eld. (2) K [x℄=hf i is an integral domain. (3) f is irredu ible.

1.1.6. An irredu ible polynomial f = an xn + + a1 x + a0 2 K [x℄, K a

eld, is alled separable , if f has only simple roots in K , the algebrai losure of K . An algebrai eld extension K L is alled separable if any element a 2 L is separable over K , that is, the minimal polynomial of a over K is separable. (1) Show that f 6= 0 is separable if and only if f and its formal derivative Df := nan xn 1 + + a1 have no ommon fa tor of degree 1. (2) A nite separable eld extension K L is generated by a primitive element , that is, there exists an irredu ible f 2 K [x℄ su h that K [x℄=hf i. L= (3) K is alled a perfe t eld if every irredu ible polynomial f 2 K [x℄ is separable. Show that nite elds, algebrai ally losed elds and elds of

hara teristi 0 are perfe t.

1.1.7. Whi h of the elds in Singular, (1) { (5), are perfe t, whi h not? 1.1.8. Compute (10!)^5 with the help of Singular. 1.1.9. De lare in Singular a polynomial ring in the variables x(1), x(2), x(3), x(4) over the nite eld with eight elements

1.1.10. De lare in Singular the ring A = Q (a; b; )[x; y; z; w℄ and ompute f 2 = 2 for f = (ax3 + by2 + z 2)(a b ). 1.1.11. De lare in Singular the rings A = Q [a; b; ℄ and B = Q [a℄. In A de ne the polynomial f = a + b + ab + 3 . Try in B the ommands imap(A,f) and fet h(A,f).

1.1.12. De lare in

3

Singular

(i + i2 + 1)(uvw) .

the ring Q (i)[u; v; w℄, i2 = 1, and ompute

1.1.13. Write a Singular pro edure, depending on two integers p; d, with

Fp [x℄ of degree d su h that the

orresponding polynomial fun tion vanishes. Use the pro edure to display all f 2 (Z=5Z)[x℄ of degree 6 su h that f~ = 0.

p a prime, whi h returns all polynomials in

1.2 Monomial Orderings The presentation of a polynomial as a linear ombination of monomials is unique only up to an order of the summands, due to the ommutativity of the addition. We an make this order unique by hoosing a total ordering on the set of monomials. For further appli ations it is ne essary, however, that the ordering is ompatible with the semigroup stru ture on Monn .

10


De nition 1.2.1. A monomial ordering or semigroup ordering is a total

(or linear) ordering > on the set of monomials Monn = fx j 2 N n g in n variables satisfying

x > x =) x x > x x for all ; ; 2 N n . We say also > is a monomial ordering on A[x1 ; : : : ; xn ℄, A any ring, meaning that > is a monomial ordering on Monn . We identify Monn with N n , and then a monomial ordering is a total ordering on N n , whi h is ompatible with the semigroup stru ture on N n given by addition. A typi al, and important, example is provided by the lexi ographi al ordering on N n : x > x if and only if the rst non{zero entry of is positive. We shall see dierent monomial orderings later. Monomial orderings provide an extra stru ture on the set of monomials and, hen e, also on the polynomial ring. Although they have been used in several pla es to prove diÆ ult mathemati al theorems they are hardly part of lassi al ommutative algebra. Monomial orderings, however, an be quite powerful tools in theoreti al investigations ( f. [79℄) but, in addition, they are indispensable in many serious and deeper polynomial omputations. From a pra ti al point of view, a monomial ordering > allows us to write a polynomial f 2 K [x℄ in a unique ordered way as

f = a x + a x + + a x ; with x > x > > x , where no oeÆ ient is zero (a sparse representation of f ). Moreover, this allows the representation of a polynomial in a omputer as an ordered list of oeÆ ients, making equality tests very simple and fast (assuming this is the ase for the ground eld). Additionally, this order does not hange if we multiply f with a monomial. For highly sophisti ated presentations of monomials and polynomials in a omputer see [8℄. There are many more and deeper properties of monomial orderings and, moreover, dierent orderings have dierent further properties.

De nition 1.2.2. Let > be a xed monomial ordering. Write f 2 K [x℄, f= 6 0, in a unique way as a sum of non{zero terms

f = a x + a x + + a x ; x > x > > x ; and a ; a ; : : : ; a (1) (2) (3) (4) (5)

2 K . We de ne:

LM(f ) := leadmonom(f):= x , the leading monomial of f , LE(f ) := leadexp(f):= , the leading exponent of f , LT(f ) := lead(f):= a x , the leading term or head of f , LC(f ) := lead oef(f):= a , the leading oeÆ ient of f tail(f ) := f lead(f)= a x + + a x , the tail of f .


11

Let us onsider an example with the lexi ographi al ordering. In Singular every polynomial belongs to a ring whi h has to be de ned rst. We de ne the ring A = Q [x; y; z ℄ together with the lexi ographi al ordering. A is the name of the ring, 0 the hara teristi of the ground eld Q , x; y; z are the names of the variables and lp de nes the lexi ographi al ordering with x > y > z , see Example 1.2.8.

SINGULAR Example 1.2.3 (leading data). ring A = 0,(x,y,z),lp; poly f = y4z3+2x2y2z2+3x5+4z4+5y2; f; //display //-> 3x5+2x2y2z2+y4z3+5y2+4z4 leadmonom(f); //leading //-> x5 leadexp(f); //leading //-> 5,0,0 lead(f); //leading //-> 3x5 lead oef(f); //leading //-> 3 f - lead(f); //tail //-> 2x2y2z2+y4z3+5y2+4z4

f in a lex-ordered way monomial exponent term

oeffi ient

The most important distin tion is between global and lo al orderings.

De nition 1.2.4. Let > be a monomial ordering on fx j 2 N n g.

(1) > is alled a global ordering if x > 1 for all 6= (0; : : : ; 0), (2) > is alled a lo al ordering if x < 1 for all 6= (0; : : : ; 0), (3) > is alled a mixed ordering if it is neither global nor lo al.

Of ourse, if we turn the ordering around by setting x >0 x if x > x , then >0 is global if and only if > is lo al. However, lo al and global (and mixed) orderings have quite dierent properties. Here are the most important

hara terizations of a global ordering.

Lemma 1.2.5. Let > be a monomial ordering, then the following onditions are equivalent:

(1) (2) (3) (4)

> is a well{ordering. xi > 1 for i = 1; : : : ; n. x > 1 for all 6= (0; : : : ; 0), that is, > is global. nat and 6= implies x > x .

The last ondition means that > is a re nement of the natural partial ordering on N n de ned by (1 ; : : : ; n ) nat ( 1 ; : : : ; n ) :() i i for all i :

12


Proof. (1) ) (2): if xi < 1 for some i, then xpi < xpi

1

< 1, yielding a set of monomials without smallest element (re all that a well{ordering is a total ordering on a set su h that ea h non{empty subset has a smallest element). 0 (2) ) (3): write x = x xj for some j and use indu tion. For (3) ) (4) let (1 ; : : : ; n ) nat ( 1 ; : : : ; n ) and 6= . Then := 2 N n r f0g, hen e x > 1 and, therefore, x = x x > x . (4) ) (1): Let M be a non{empty set of monomials. By Di kson's Lemma (Lemma 1.2.6) there is a nite subset B M su h that for ea h x 2 M there is an x 2 B with nat . By assumption, x < x or x = x , that is, B ontains a smallest element of M with respe t to >.

Lemma 1.2.6 (Di kson, 1913). Let M N n be any subset. Then there is a nite set B M satisfying

8 2 M 9 2 B su h that nat : B is sometimes alled a Di kson basis of M. Proof. We write instead of nat and use indu tion on n. For n = 1 we an take the minimum of M as the only element of B . For n > 1 and i 2 N de ne Mi = f0 = (1 ; : : : ; n 1 ) 2 N n 1 j (0 ; i) 2 M g and, by indu tion, Mi has a Di kson S basis Bi . Again, by indu tion hypothesis, i2N Bi has a Di kson basis B 0 . B 0 is nite, hen e B 0 B1 [ [ Bs for some s. We laim that

B := f( 0 ; i) 2 N n j 0 i s; 0 2 Bi g is a Di kson basis of M . To see this, let (0 ; n ) 2 M . Then 0 2 Mn and, sin e Bn is a Di kson basis of Mn , there is a 0 2 Bn with 0 0 . If n s, then ( 0 ; n ) 2 B and ( 0 ; n ) (0 ; n ). If n > s, we an nd a 0 2 B 0 and an i s su h that 0 0 and ( 0 ; i) 2 Bi . Then ( 0 ; i) 2 B and ( 0 ; i) (0 ; n ).

Remark 1.2.7. If A is an n n integer matrix with only non{negative entries and determinant 6= 0, and if > is a monomial ordering, we an de ne a matrix ordering >(A;>) by setting x >(A;>) x :() xA > xA where and are onsidered as olumn ve tors. By Exer ise 1.2.6 (2), >(A;>) is again a monomial ordering. We an even use matri es A 2 GL(n; R) with real entries to obtain a monomial ordering by setting

x >A x :() A > A ;


13

where > on the right{hand side is the lexi ographi al ordering on Rn . Robbiano proved in [152℄, that every monomial ordering arises in this way from the lexi ographi al ordering on Rn . However, we do not need this fa t ( f. Exer ise 1.2.9). Important examples of monomial orderings are:

Example 1.2.8 (monomial orderings).

In the following examples we x an enumeration x1 ; : : : ; xn of the variables, any other enumeration leads to a dierent ordering.

(1)

Global Orderings

(i) Lexi ographi al ordering >lp (also denoted by lex):

x >lp x :()

9 1 i n : 1 = 1 ; : : : ; i

1

= i 1 ; i > i :

(ii) Degree reverse lexi ographi al ordering >dp (denoted by degrevlex):

x >dp x :() deg x > deg x or deg x = deg x and 9 1 i n : n = n ; : : : ; i+1 = i+1 ; i < i ; where deg x = 1 + + n .

(iii) Degree lexi ographi al ordering >Dp (also denoted by deglex):

x >Dp x :() deg x > deg x or deg x = deg x and 9 1 i n : 1 = 1 ; : : : ; i 1 = i 1 ; i > i : In all three ases x1 ; : : : ; xn > 1. For example, we have x31 >lp x21 x22 but x21 x22 >dp;Dp x31 . An example where dp and Dp dier: x21 x2 x23 >Dp x1 x32 x3 but x1 x32 x3 >dp x21 x2 x23 . Given a ve tor w = (w1 ; : : : ; wn ) of integers, we de ne the weighted degree of x by w{deg(x ) := hw; i := w1 1 + + wn n ; that is, the variable xi has degree wi . For a polynomial f = we de ne the weighted degree,

P

a x ,

w{deg(f ) := max w{deg(x ) a 6= 0 : Using the weighted degree in (ii), respe tively (iii), with all wi > 0, instead of the usual degree, we obtain the weighted reverse lexi ographi al ordering, wp(w1 ; : : : ; wn ), respe tively the weighted lexi ographi al ordering, Wp(w1 ; : : : ; wn ).

14

(2)

1. Rings, Ideals and Standard Bases Lo al Orderings

(i) Negative lexi ographi al ordering >ls :

x >ls x :()

9 1 i n; 1 = 1 ; : : : ; i

1

= i 1 ; i < i :

(ii) Negative degree reverse lexi ographi al ordering >ds :

x >ds x :() deg x < deg x ; where deg x = 1 + + n ; or deg x = deg x and 9 1 i n : n = n ; : : : ; i+1 = i+1 ; i < i : (iii) Negative degree lexi ographi al ordering >Ds :

x >Ds x :() deg x < deg x ; or deg x = deg x and 9 1 i n : 1 = 1 ; : : : ; i 1 = i 1 ; i > i : Similarly, as above, we an de ne weighted versions ws(w1 ; : : : ; wn ) and Ws(w1 ; : : : ; wn ) of the two last lo al orderings. (3)

Produ t or Blo k Orderings

Now onsider >1 , a monomial ordering on Mon(x1 ; : : : ; xn ), and >2 , a monomial ordering on Mon(y1 ; : : : ; ym ). Then the produ t ordering or blo k ordering >, also denoted by (>1 ; >2) on Mon(x1 ; : : : ; xn ; y1 ; : : : ; ym), is de ned as 0

0

0

x y > x y :() x >1 x 0 0 or x = x and y >2 y : If >1 is a global ordering then the produ t ordering has the property that monomials whi h ontain an xi are always larger than monomials ontaining no xi . If the spe ial orderings >1 on Mon(x1 ; : : : ; xn ) and >2 on Mon(y1 ; : : : ; ym ) are irrelevant, for a produ t ordering on Mon(x1 ; : : : ; xn ; y1 ; : : : ; ym ) we write just x y . If >1 and >2 are global (respe tively lo al), then the produ t ordering is global (respe tively lo al) but the produ t ordering is mixed if one of the orderings >1 and >2 is global and the other lo al. This is how mixed orderings arise in a natural way.

De nition 1.2.9. A monomial ordering > on fx j 2 N n g is alled a

weighted degree ordering if there exists a ve tor w = (w1 ; : : : ; wn ) of non{ zero integers su h that

w{deg(x ) > w{deg(x ) =) x > x : It is alled a global (respe tively lo al ) degree ordering if the above holds for wi = 1 (respe tively wi = 1) for all i.


15

Remark 1.2.10. Consider a matrix ordering de ned by A 2 GL(n; R). Sin e the olumns of A are lexi ographi ally greater than the 0{ve tor if and only if the variables are greater than 1, it follows that a matrix ordering >A is a well{ordering if and only if the rst non{zero entry in ea h olumn of A is

positive. It is a (weighted) degree ordering if and only if all entries in the rst row of A are non{zero. Of ourse, dierent matri es an de ne the same ordering. For examples of matri es de ning the above orderings see the Singular Manual.

Although we an represent any monomial ordering > as a matrix ordering >A for some A 2 GL(n; R), it turns out to be useful to represent > just by one weight ve tor. This is, in general, not possible on the set of all monomials ( f. Exer ise 1.2.10) but it is possible, as we shall see, for nite subsets. For this purpose, we introdu e the set of dieren es

D := f j x > x g Zn asso iated to a monomial ordering on Mon(x1 ; : : : ; xn ). D has the following properties,

0 26 D, 1 ; 2 2 D =) 1 + 2 2 D. The last property follows from the fa t that > is a semigroup ordering. Namely, if 1 = 1 1 , 2 = 2 2 2 D, then x1 > x 1 implies that x1 +2 > x 1 +2 , and x2 > x 2 implies that x 1 +2 > x 1 + 2 , therefore x1 +2 > x 1 + 2 and

+ 2 = (1 + 2 ) ( 1 + 2 ) 2 D. Pk 1 It follows that i=1 ni i 2 D for ni 2 N r f0g and i 2 D, and, hen e, Pk of elements of D with ri 2 Q >0 . i=1 ri i 6= 0 for any nite linear ombination Pk P In parti ular, no onvex ombination i=1 ri i , ri 2 Q 0 , ki=1 ri = 1, yields 0, that is, 0 is not ontained in the onvex hull of D. This fa t will be used in the following lemma.

Lemma 1.2.11. Let > be a monomial ordering and M Mon(x1 ; : : : ; xn ) a nite set. Then there exists some w = (w1 ; : : : ; wn ) 2 Zn su h that x > x if and only if hw; i > hw; i for all x ; x 2 M. Moreover, w an be hosen su h that wi > 0 for xi > 1 and wi < 0 if xi < 1. The integer ve tor w is alled a weight{ve tor and we say that w indu es > on M .

Proof. Sin e hw; i > hw; i if and only if hw; w 2 Zn su h that hw; i > 0 for all

i > 0, we have to nd

2 DM := f 2 D j x ; x 2 M; x > x g : This means that DM should be in the positive half{spa e de ned by the linear form hw; i on Q n . Sin e 0 is not ontained in the onvex hull of DM and

16


sin e DM is nite, we an, indeed, nd su h a linear form (see, for example, [174℄, Theorem 2.10). To see the last statement, in lude 1 and xi , i = 1; : : : ; n, into M . Then wi > 0 if xi > 1 and wi < 0 if xi < 1.

Example 1.2.12. A weight ve tor for the lexi ographi al ordering lp an be determined as follows. For M Monn nite, onsider an n{dimensional ube spanned by the oordinate axes ontaining M . Choose an integer v larger than the side length of this ube. Then w = (v n 1 ; v n 2 ; : : : ; v; 1) indu es lp on M. We shall now de ne in Singular the same ring Q [x; y; z ℄ with dierent orderings, whi h are onsidered as dierent rings in Singular. Then we map a given polynomial f to the dierent rings using imap and display f as a sum of terms in de reasing order, the method by whi h f is represented in the given ring.

SINGULAR Example 1.2.13 (monomial orderings).

Global orderings are denoted with a p at the end, referring to \polynomial ring" while lo al orderings end with an s, referring to \series ring". Note that Singular stores and outputs a polynomial in an ordered way, in de reasing order. (1) Global orderings: ring A1 = 0,(x,y,z),lp; //lexi ographi al poly f = x3yz + y5 + z4 + x3 + xy2; f; //-> x3yz+x3+xy2+y5+z4 ring A2 = 0,(x,y,z),dp; poly f = imap(A1,f); f; //-> y5+x3yz+z4+x3+xy2

//degree reverse lexi ographi al

ring A3 = 0,(x,y,z),Dp; poly f = imap(A1,f); f; //-> x3yz+y5+z4+x3+xy2

//degree lexi ographi al

ring A4 = 0,(x,y,z),Wp(5,3,2);//weighted degree //lexi ographi al poly f = imap(A1,f); f; //-> x3yz+x3+y5+xy2+z4

(2) Lo al orderings: ring A5 = 0,(x,y,z),ls; poly f = imap(A1,f); f; //-> z4+y5+xy2+x3+x3yz

//negative lexi ographi al

1.2 Monomial Orderings ring A6 = 0,(x,y,z),ds; poly f = imap(A1,f); f; //-> x3+xy2+z4+y5+x3yz

17

//negative degree reverse //lexi ographi al

ring A7 = 0,(x,y,z),Ws(5,3,2);//negative weighted degree //lexi ographi al poly f = imap(A1,f); f; //-> z4+xy2+x3+y5+x3yz

(3) Produ t and matrix orderings: ring A8 = 0,(x,y,z),(dp(1),ds(2)); //mixed produ t ordering poly f = imap(A1,f); f; //-> x3+x3yz+xy2+z4+y5 intmat A[3℄[3℄ print(A); //-> -1 //-> 0 //-> 0

= -1, -1, -1, 0, 0, 1, 0, 1, 0; -1 0 1

-1 1 0

Now de ne your own matrix ordering using A: ring A9 = 0,(x,y,z),M(A); //a lo al ordering poly f = imap(A1,f); f; //-> xy2+x3+z4+x3yz+y5

Exer ises 1.2.1. Show that lp, dp, Dp, wp(w(1..m)), Wp(w(1..n)), respe tively ls, ds, Ds, ws(w(1..m)), Ws(w(1..n)), as de ned in Example 1.2.8 are indeed

global, respe tively lo al, monomial orderings. 1.2.2. Determine the names of the orderings given by the following matri es: 1 1; 0 1

( 10 01 ) ;

1 1; 0 1

1 0; 0 1

1 2; 0 1

1 0 0 0

1 1 0 0

0 0 1 0

0 0 : 1 1

1.2.3. Order the polynomial x4 + z 5 + x3 z + yz 4 + x2 y2 with respe t to the

orderings dp,Dp,lp,ds,Ds,ls,wp(5,3,4),ws(5,5,4). 1.2.4. Compute the leading term and the leading oeÆ ient

f = 4xy2z + 4z 2

5x3 + 7xy 2

7y 4

with respe t to the orderings lp on Q [x; y; z ℄, lp on Q (x)[z; y ℄, lp on Q [z; y; x℄, Dp on (Z=2Z)[z; y; x℄, ls on Q [x; y; z ℄, wp(w(1..3)) on (Z=2Z)[x; y; z ℄, where wp(w(1..3)) is given by w{deg(x y z ) := 3 + 2 + .

18


1.2.5. Determine matri es de ning the orderings wp(5,3,4), ws(5,5,4).

dp, Dp, lp, ds, Ds, ls,

1.2.6. Let > be any monomial ordering on Mon(x1 ; : : : ; xn ). (1) Let w = (w1 ; : : : ; wn ) 2 Rn be arbitrary. Show that

x >w x :() hw; i > hw; i or hw; i = hw; i and x > x de nes a monomial ordering on Mon(x1 ; : : : ; xn ). Note that the ordering >w is a (weighted) degree ordering. It is a global ordering if wi > 0 for all i and a lo al ordering if wi < 0 for all i. (2) Let A be an n n integer matrix with non{negative entries, whi h is invertible over Q . Show that

x >(A;>) x , xA > xA de nes a monomial ordering on Mon(x1 ; : : : ; xn ).

1.2.7. (1) Prove the laim made in Example 1.2.12.

(2) Consider a matrix ordering >A for some matrix A 2 GL(n; Q ) and M Monn a nite set. Use (1) and the fa t that x >A x if and only if A >lex A to determine a weight ve tor whi h indu es >A on M .

1.2.8. (1) Determine weight ve tors w whi h indu e dp, respe tively ds, on

M = fxi yj z k j 1 i; j; k 5g. (2) Che k your result, using Singular , in the following way: reate a polynomial f , being the sum of all monomials of degree 5 in the rings with ordering dp, respe tively ds, and onvert f to a string. Then do the same in the rings with ordering wp(w), respe tively ws(-w), ((a(w),lp), respe tively (a(-w),lp)), and ompare the respe tive strings.

1.2.9. Show that any monomial ordering > an be de ned as >A by a matrix A 2 GL(n; R).

(Hint: You may pro eed as follows: rst show that a semigroup ordering on (Zn0; +) extends in a unique way to a group ordering on (Q n ; +). Then show that, for any Q {subve tor spa e V Q n of dimension r, the set (

V0 :=

> 0 9 z+ ("); z (") 2 U" (z ) \ V z 2 R 8 "su h that z+ (") > 0; z (") < 0 n

is an R{subve tor spa e in Rn of dimension r su

essively, the rows of A.)

)

1. Use this to onstru t,

1.2.10. Let w1 ; : : : ; wP independent over Q and de ne > n 2 R be linearly P

by setting x < x if ni=1 wi i < ni=1 wi i . Prove that > is a monomial ordering. Show that there is no matrix A 2 GL(n; Q ) de ning this ordering.

1.3 Ideals and Quotient Rings


19

Ideals are in the entre of ommutative algebra and algebrai geometry. Here we introdu e only the basi notions related to them. Let A be a ring, as always, ommutative and with 1.

De nition 1.3.1. A subset I A is alled an ideal if it is an additive subgroup whi h is losed under s alar multipli ation, that is, f; g 2 I =) f + g 2 I f 2 I; a 2 A =) af 2 I:

De nition 1.3.2. (1) Let I A be an ideal. A family (f )2 , any index set, and f 2 I , is

alled a system of generators of I ifPevery element f 2 I an be expressed as a nite linear ombination f = a f for suitable a 2 A. We then write

I = hf j 2 iA = hf j 2 i = or, if = f1; : : : ; k g,

X

2

f A

I = hf1 ; : : : ; fk iA = hf1 ; : : : ; fk i : (2) I is alled nitely generated if it has a nite system of generators; it is

alled prin ipal if it an be generated P by one element. (3) If (IS )2 is a family of ideals, then 2 I denotes the ideal generated by 2 I . (4) If I1 ; I2 are ideals, then I1 I2 (or I1 I2 ) denotes the ideal generated by the set fab j a 2 I1 ; b 2 I2 g. Note that the union of ideals is, in general, not an ideal (but the interse tion is). We have X

2

(

I =

X

2

a a

2 I ; a = 0 for almost all

)

:

Be ause the empty sum is de ned to be 0, the 0{ideal is generated by the P empty set (but also by 0). The expression f = a f as a linear ombination of the generators is, in general, by no means unique. For example, if I = hf1 ; f2 i then we have the trivial relation f1 f2 f2 f1 = 0, hen e a1 f1 = a2 f2 with a1 = f2 , a2 = f1 . Usually there are also further relations, whi h lead to the notion of the module of syzygies ( f. Chapter 2). Ideals o

ur in onne tion with ring maps. If ' : A ! B is a ring homomorphism and J B an ideal, then the preimage

20


' 1 (J ) = fa 2 A j '(a) 2 J g is an ideal. In parti ular, Ker ' = fa 2 A j '(a) = 0g is an ideal in A. On the other hand, the image

'(I ) = f'(a) j a 2 I g of an ideal I A is, in general, not an ideal. In parti ular, Im ' = '(A) B is not, generally, an ideal (for example, onsider Z Q , then no non{zero ideal in Z is an ideal in Q ). All these statements are very easy to he k. ' is alled inje tive if Ker ' = 0, and surje tive if Im ' = B . A bije tive , that is inje tive and surje tive, morphism is alled an isomorphism , an isomorphism from A to A an automorphism .

ontains the built{in ommand preimage whi h an be used to

ompute the kernel of a ring map. If a ring map ' : K [x1 ; : : : ; xk ℄ ! K [y1 ; : : : ; ym ℄ is given by f1 ; : : : ; fk , that is, '(xi ) = fi , then ' is surje tive if and only if y1 ; : : : ; ym are ontained in the subring Im ' = K [f1 ; : : : ; fm ℄ of K [y1 ; : : : ; ym ℄. This fa t is used in Singular to he k surje tivity. We shall explain the algorithms for he king inje tivity, surje tivity, bije tivity of a ring map in Chapter 2. Here we just apply the orresponding pro edures from algebra.lib.

Singular

SINGULAR Example 1.3.3 (properties of ring maps).

(1) Che king inje tivity:

ring S = 0,(a,b, ),lp; ring R = 0,(x,y,z),dp; ideal i = x, y, x2-y3; map phi = S,i; //a map from S to R, a->x, b->y, ->x2-y3 LIB "algebra.lib"; //load algebra.lib

By default, Singular displays the names and paths of those libraries whi h are used by algebra.lib and whi h are also loaded. We suppress this message. We test inje tivity using the pro edure is_inje tive, then we ompute the kernel by using the pro edure alg_kernel (whi h displays the kernel, an obje t of the preimage ring, as a string). is_inje tive(phi,S); //-> 0

// phi is not inje tive

1.3 Ideals and Quotient Rings ideal j = x, x+y, z-x2+y3; map psi = S,j; is_inje tive(psi,S); //-> 1

21

// another map from S to R // psi is inje tive

alg_kernel(phi,S); //-> b^3-a^2+ alg_kernel(psi,S); //-> 0

// = Ker(phi)

(2) Computing the preimage : Using the preimage ommand, we must rst go ba k to S, sin e the preimage is an ideal in the preimage ring. ideal Z; setring S; preimage(R,phi,Z); //-> _[1℄=a2-b3-

//the zero ideal in R // omputes kernel of phi in S //kernel of phi = preimage of Z

(3) Che king surje tivity and bije tivity . setring R; is_surje tive(psi,S); //-> 1 is_bije tive(psi,S); //-> 1

//faster than is_inje tive, //is_surje tive

De nition 1.3.4. A ring A is alled Noetherian if every ideal in A is nitely

generated.

It is a fundamental fa t that the polynomial ring A[x1 ; : : : ; xn ℄ over a Noetherian ring A is again Noetherian; this is the ontent of the Hilbert basis theorem. Sin e a eld is obviously a Noetherian ring, the polynomial ring over a eld is Noetherian. It follows that the kernel of a ring map between Noetherian rings is nitely generated. An important point of the Singular Example 1.3.3 is that we an expli itly ompute a nite set of generators for the kernel of a map between polynomial rings.

Theorem 1.3.5 (Hilbert basis theorem). If A is a Noetherian ring then the polynomial ring A[x1 ; : : : ; xn ℄ is Noetherian.

22


For the proof of the Hilbert basis theorem we use

Proposition 1.3.6. The following properties of a ring A are equivalent: (1) A is Noetherian. (2) Every as ending hain of ideals I1 I2 I3 : : : Ik : : : be omes stationary (that is, there exists some j0 su h that Ij = Ij0 for all j j0 ). (3) Every non{empty set of ideals in A has a maximal element (with regard to in lusion). Condition (2) is alled the as ending hain ondition and (3) the maximality

ondition . We leave the proof of this proposition as Exer ise 1.3.9.

Proof of Theorem 1.3.5. We need to show the theorem only for n = 1, the general ase follows by indu tion. We argue by ontradi tion. Let us assume that there exists an ideal I A[x℄ whi h is not nitely generated. Choose polynomials

f1 2 I; f2 2 I r hf1 i; : : : ; fk+1 2 I r hf1 ; : : : ; fk i; : : : of minimal possible degree. If di = deg(fi ),

fi = ai xdi + lower terms in x ; then d1 d2 : : : and ha1 i ha1 ; a2 i : : : is an as ending hain of ideals in A. By assumption it is stationary, that is, ha1 ; : : : ; ak i = ha1 ; : : : ; ak+1 i for P some k , hen e, ak+1 = ki=1 bi ai for suitable bi 2 A. Consider the polynomial

g = fk+1

k X i=1

bi xdk+1 di fi = ak+1 xdk+1

k X i=1

bi ai xdk+1 + lower terms :

Sin e fk+1 2 I rhf1 ; : : : ; fk i, it follows that g 2 I rhf1 ; : : : ; fk i is a polynomial of degree smaller than dk+1 , a ontradi tion to the hoi e of fk+1 .

De nition 1.3.7. Let I be any ideal in the ring A. We de ne the quotient ring or fa tor ring A=I as follows.

(1) A=I is the set of o{sets f[a℄ := a + I j a 2 Ag2 with addition and multipli ation de ned via representatives:

2 a + I := fa + f

j f 2 I g.

[a℄ + [b℄ := [a + b℄; [a℄ [b℄ := [a b℄:


23

It is easy to see that the de nitions are independent of the hosen representatives and that (A=I; +; ) is, indeed, a ring. Moreover, A=I is not the zero ring if and only if 1 62 I . (2) The residue map or quotient map is de ned by

:A

! A=I ;

a7

! [a℄ :

is a surje tive ring homomorphism with kernel I . The following lemma is left as an easy exer ise.

Lemma 1.3.8. The map J 7! (J ) indu es a bije tion

fideals in A ontaining I g ! fideals in A=I g

with J 0 7! 1 (J 0 ) being the inverse map.

De nition 1.3.9.

(1) An element a 2 A is alled a zerodivisor if there exists an element b 2 A r f0g satisfying ab = 0; otherwise a is a non{zerodivisor . (2) A is alled an integral domain if A 6= 0 and if A has no zerodivisors ex ept 0. (3) A is a prin ipal ideal ring if every ideal in A is prin ipal; if A is, moreover, an integral domain it is alled a prin ipal ideal domain . Polynomial rings over a eld are integral domains (Exer ise 1.3.1 (4)). This is, however, not generally true for quotient rings K [x1 ; : : : ; xn ℄=I . For example, if I = hf g i with f; g 2 K [x1 ; : : : ; xn ℄ polynomials of positive degree, then [f ℄ and [g ℄ are zerodivisors in K [x1 ; : : : ; xn ℄=I and not zero. A ring A, whi h is isomorphi to a fa tor ring K [x1 ; : : : ; xn ℄=I , is alled an aÆne ring over K .

De nition 1.3.10. Let I A be an ideal.

(1) I is a prime ideal if I 6= A and if for ea h a; b 2 A : ab 2 I ) a 2 I or b 2 I. (2) I is a maximal ideal if I 6= A and if it is maximal with respe t to in lusion (that is, for any ideal I 0 ( A and I I 0 implies I = I 0 ). (3) The set of prime ideals is denoted by Spe (A) and the set of maximal ideals by Max(A). The set of prime ideals Spe (A) of a ring A is made a topologi al spa e by endowing it with the so{ alled Zariski topology, reating, thus, a bridge between algebra and topology. We refer to the Appendix, in parti ular A.3, for a short introdu tion. In many ases in the text we use Spe (A) just as a set. But, from time to time, when we think we should relax and enjoy geometry, then we onsider the aÆne spa e Spe (A) instead of the ring A and the variety V (I ) Spe (A) instead of the ideal I . Most of the examples deal with aÆne rings over a eld K .

24


Lemma 1.3.11. (1) I A is a prime ideal if and only if A=I is an integral domain. (2) I A is a maximal ideal if and only if A=I is a eld. (3) Every maximal ideal is prime. Proof. Let I ( A. For a; b 2 A we have ab 2 I () [ab℄ = [a℄ [b℄ = 0 in A=I , whi h implies (1). By Lemma 1.3.8, A=I has only the trivial ideals 0 and A=I , if and only if I and A are the only ideals of A whi h ontain I , whi h

implies (2). Finally, (3) follows from (2) and (1), sin e a eld is an integral domain.

If ' : A ! B is a ring map and I B is a prime ideal, then ' 1 (I ) is a prime ideal (an easy he k). However, the preimage of a maximal ideal need not be maximal. (Consider Z Q , then 0 is a maximal ideal in Q but not in Z.)

Lemma 1.3.12. Let A be a ring. (1) Let P; I; J A be ideals with P prime. Then I 6 P , IJ P implies J P. T (2) Let I1T ; : : : ; In , P A be ideals with P prime and ni=1 Ii P (respe tively i Ii = P ), then P Ii (respe tively P = Ii ) for some i. (3) ( Prime S avoidan e) Let P1 ; : : : ; Pn , I A be ideals with Pi prime and I ni=1 Pi , then I Pi for some i.

Proof. For (1) let J = hf1 ; : : : ; fn i and x 2 I su h that x 62 P . By assumption, we have xfi 2 P for all i. Now P is prime and, therefore, fi 2 P for all i. This implies that J P . T Q Ii P and, therefore, using To prove (2) assume that Ii P . Then T (1), Ik P for some k . If, additionally, Ii = P , then P = Ik . To prove (3) we use indu tion on n. SThe ase n = 1 is trivial. Assume (3) is true for n 1 prime ideals. If I j = 6 i Pj for some i, then I Pk for some k . S We may assume now that S I 6 j= 6 i Pj for all i = 1; : : : ; n and hoose P . This implies espe ially that xi 2 Pi x1 ; : : : ; xn 2 I su h that x 2 6 j i j 6 = i S be ause xi 2 I Pj . S Now onsider the element x1 + x2 : : : xn 2 I . Sin e I Pj , there exists a k su h that x1 + x2 : : : xn 2 Pk . If k = 1 then, sin e x1 2 P1 , we obtain x2 : : : xn 2 P1 . This implies that S x` 2 P1 for some ` > 1 whi h is a ontradi tion to the hoi e of x` 62 j = 6 ` Pj . If k > 1 then, sin e x2 : : : x n 2 P , we obtain x 2 P whi h is again a ontradi tion to the 1 k Sk

hoi e of x1 62 j = P . 6 1 j Many of the on epts introdu ed so far in this se tion an be treated ee tively using Singular. We de ne a quotient ring and test equality and the zerodivisor property in the quotient ring.


25

SINGULAR Example 1.3.13 ( omputation in quotient rings).

(1) De ne a quotient ring :

ring R = 32003,(x,y,z),dp; ideal I = x2+y2-z5, z-x-y2; qring Q = groebner(I); //defines the quotient ring Q = R/I Q; //-> // hara teristi : 32003 //-> // number of vars : 3 //-> // blo k 1 : ordering dp //-> // : names x y z //-> // blo k 2 : ordering C //-> // quotient ring from ideal //-> _[1℄=y2+x-z //-> _[2℄=z5-x2+x-z

(2) Equality test in quotient rings: Equality test in quotient rings is diÆ ult. The test f==g he ks only formal equality of polynomials, it does not work orre tly in quotient rings. Instead, we have to ompute a normal form of the dieren e f g. Why and how this works, will be explained in Se tion 1.6 on standard bases. poly f = z2 + y2; poly g = z2+2x-2z-3z5+3x2+6y2; redu e(f-g,std(0)); //normal form, result is 0 iff f=g in Q //-> 0

The same an be tested without going to the quotient ring. setring R; poly f = z2 + y2; poly g = z2 + 2x - 2z - 3z5 + 3x2 + 6y2; redu e(f-g,groebner(I)); //result is 0 iff f-g is in I //-> 0

(3) Zerodivisor test in quotient rings: setring Q; ideal q = quotient(0,f);//this defines q = : q = redu e(q,std(0)); //normal form of ideal q in Q size(q); //the number of non-zero generators //-> 0 //hen e, f is a non-zerodivisor in Q

Testing primality of a prin ipal ideal hf i in a polynomial ring is easily a hieved by using fa torize(f);. For an arbitrary ideal this is mu h more involved. One an use primde GTZ or primde SY from primde .lib, as will

26


be explained in Chapter 4. (4) Computing the inverse in quotient rings: If I K [x℄ = K [x1 ; : : : ; xn ℄ is a maximal ideal, then the quotient ring K [x℄=I is a eld. To be able to ompute ee tively in the eld K [x℄=I we need, in addition to the ring operations, the inverse of a non{zero element. The following example shows that we an ee tively ompute in all elds of nite type over a prime eld. If the polynomial f is invertible, then the ommand lift(f,1)[1,1℄ gives the inverse (lift he ks whether 1 2 hf i and then expresses 1 as a multiple of f ): ring R=(0,x),(y,z),dp; ideal I=-z5+y2+(x2),-y2+z+(-x); I=std(I); qring Q=I;

We shall now ompute the inverse of z in Q = R=I . poly p=lift(z,1)[1,1℄; p; //->1/(x2-x)*z4-1/(x2-x)

We make a test for p being the inverse of z . redu e(p*z,std(0)); //->1

The ideal I is a maximal ideal if and only if R=I is a eld. We shall now prove that, in our example, I is a maximal ideal. ring R1=(0,x),(z,y),lp; ideal I=imap(R,I); I=std(I); I; //-> I[1℄=y10+(5x)*y8+(10x2)*y6+(10x3)*y4+(5x4-1)*y2+(x5-x2) //-> I[2℄=z-y2+(-x)

Sin e Q (x)[z; y ℄=hz

y2 xi = Q (x)[y ℄, we see that

R=I = Q (x)[y ℄=hy 10 + 5xy 8 + 10x2 y 6 + 10x3 y 4 + (5x4 1)y 2 + x5 x2 i : fa torize(I[1℄); //-> [1℄: //-> _[1℄=1 //-> _[2℄=y10+(5x)*y8+(10x2)*y6+(10x3)*y4+(5x4-1)*y2 //-> +(x5-x2) //-> [2℄: //-> 1,1


27

The polynomial is irredu ible and, therefore, R=I is a eld and I a maximal ideal.

De nition 1.3.14. Let A be a ring and I; J A ideals. (1) The ideal quotient of I by J is de ned as

I : J := a 2 A aJ

I

:

The saturation of I with respe t to J is

I : J 1 = a 2 A 9 n su h that aJ n I : (2) The radi al of I , denoted by

p

I = a2

p

A

I or rad(I ) is the ideal

9 d 2 N su h that ad 2 I

;

whi h p is an ideal ontaining I . I is alled redu ed or a radi al ideal if I = I. (3) a 2 A is alled nilpotent if an = 0 for some n 2 N ; the minimal n is alled p index of nilpoten y . The set of nilpotent elements of A is equal to h0i and alled the nilradi al of A. (4) The ring A itself p is alled redu ed if it has no nilpotent elements ex ept 0, that is, if

h0i = h0i. For any ring, the quotient ring p Ared = A= h0i

is alled the redu tion of A or the redu ed ring asso iated to A. The ideal quotient I : J is an ideal in A whi h is very useful. In Singular the

ommand quotient(I,J); omputes generators of this ideal. In parti ular,

h0i : J = AnnA (J ) of J and, hen e, h0i : hf i = h0i if and only if f

is the annihilator is a non{ zerodivisor of A. It is lear that Ared is redu ed and that A = Ared if and only if A is redu ed. Any integral domain is redu ed. Computing the radi al is already quite involved ( f. Chapter 4). The radi al membership problem is, however, mu h easier ( f. Se tion 1.8.6).

SINGULAR Example 1.3.15 ( omputing with radi als).

(1) Compute the radi al of an ideal: ring R = 0,(x,y,z),dp; poly p = z4+2z2+1; LIB "primde .lib";

//loads library for radi al

28

1. Rings, Ideals and Standard Bases radi al(p); //-> _[1℄=z2+1

//squarefree part of p

ideal I = xyz, x2, y4+y5; //a more ompli ated ideal radi al(I); //-> _[1℄=x //-> _[2℄=y2+y //we see that I is not redu ed

(2) Compute the index of nilpoten y in a quotient ring: Sin e y 2 + y is ontained in the radi al of I , some power of y 2 + y must be

ontained in I . We ompute the minimal power k so that (y 2 + y )k is ontained in I by using the normal form as in Example 1.3.13. This is the same as saying that y 2 + y is nilpotent in the quotient ring R=I and then k is the index of nilpoten y of y 2 + y in R=I . ideal Is = groebner(I); int k; while (redu e((y2+y)^k,Is) != 0 ) {k++;} k; //-> 4 //minimal power (index of nilpoten y) is 4

Exer ises

P

1.3.1. Let A be a ring and f = jj0 a x 2 A[x1 ; : : : ; xn ℄. Prove the fol-

lowing statements:

(1) f is nilpotent if and only if a is nilpotent for all . (Hint: hoose a monomial ordering and argue by indu tion on the number of summands.) In parti ular: A[x1 ; : : : ; xn ℄ is redu ed if and only if A is redu ed. (2) f is a unit in A[x1 ; : : : ; xn ℄ if and only if a0;:::;0 is a unit in A and a are nilpotent for 6= 0. (Hint: Remember the geometri series for 1=(1 g ) and use (1).) In parti ular: (A[x1 ; : : : ; xn ℄) = A if and only if A is redu ed. (3) f is a zerodivisor in A[x1 ; : : : ; xn ℄ if and only if there exists some a 6= 0 in A su h that af = 0. Give two proofs: one by indu tion on n, the other by using a monomial ordering. (Hint: hoose a monomial ordering and g 2 A[x1 ; : : : ; xn ℄ with minimal number of terms so that f g = 0, onsider the biggest term and on lude that g must be a monomial.) (4) A[x1 ; : : : ; xn ℄ is an integral domain if and only if deg(fg ) = deg(f ) + deg(g ) for all f; g 2 A[x1 ; : : : ; xn ℄. In parti ular: A[x1 ; : : : ; xn ℄ is an integral domain if and only if A is an integral domain.


29

1.3.2. Let ' : A ! B be a ring homomorphism, I an ideal in A and J an ideal in B . Show that:

(1) ' 1 (J ) I is an ideal. (2) '(I ) is a subring of B , not ne essarily with 1, but, in general, not an ideal. (3) If ' is surje tive then '(I ) is an ideal in B .

1.3.3. Prove the following statements: (1)

Z and the polynomial ring K [x℄ in one variable over a eld are prin ipal

ideal domains [use division with remainder℄. (2) Let A be any ring, then A[x1 ; : : : ; xn ℄, n > 1, is not a prin ipal ideal domain.

1.3.4. Let A be a ring. A non{unit f 2 A is alled irredu ible if f = f1 f2 ,

f1 ; f2 2 A, implies that f1 or f2 is a unit. f is alled a prime element if hf i is a prime ideal. Prove Exer ise 1.1.5 with K [x℄ repla ed by any prin ipal ideal domain A. Moreover, prove that the onditions (1) { (3) of Exer ise 1.1.5 are equivalent to

(4) The ideal hf i is a prime ideal. (5) The ideal hf i is a maximal ideal.

1.3.5. Let R be a prin ipal ideal domain. Use Exer ise 1.3.4 to prove that every non{unit f 2 R an be written in a unique way as a produ t of nitely many prime elements. Unique means here modulo permutation and multipli ation with a unit. 1.3.6. The quotient ring of a prin ipal ideal ring is a prin ipal ideal ring.

Show, by an example, that the quotient ring of an integral domain (respe tively a redu ed ring) need not be an integral domain.

1.3.7. (1) If A; B are prin ipal ideal rings, then, also A B .

(2) A B is never an integral domain, unless A or B are trivial. (3) How many ideals has K F if K and F are elds?

1.3.8. Prove the following statements:

(1) Let n > 1, then Z=nZ is redu ed if and only if n is a produ t of pairwise dierent primes. (2) Let K be a eld, and let f 2 K [x1 ; : : : ; xn ℄ be a polynomial of degree 1. Then K [x1 ; : : : ; xn ℄=hf i is redu ed (respe tively an integral domain) if and only if f is a produ t of pairwise dierent irredu ible polynomials (respe tively irredu ible).

1.3.9. Prove Proposition 1.3.6. 1.3.10. Prove Lemma 1.3.8. 1.3.11. Let A be a Noetherian ring, and let I A be an ideal. Prove that A=I is Noetherian.

30


1.3.12. Let A be a Noetherian ring, and let ' : A ! A be a surje tive ring homomorphism. Prove that ' is inje tive.

1.3.13. (Chinese remainder T theorem ) Let A be a ring, and let I1 ; : : : ; Is be ideals in A. Assume that sj=1 Ij = h0i and Ij + Ik = A for j 6= k . Prove that the anoni al map

A

!

s M j =1

A=Ij ; a 7

! (a + I1 ; : : : ; a + Is ) ;

is an isomorphism of rings.

1.3.14. Let K be a eld and A a K {algebra. Then A is alled an Artinian K{algebra if dimK (A) < 1. Prove the following statements: (1) An Artinian K {algebra is Noetherian. (2) A is an Artinian K {algebra if and only if ea h des ending hain of ideals I1 I2 I3 : : : Ik : : : be omes stationary (that is, there exists some j0 su h that Ij = Ij0 for all j j0 ). 3

1.3.15. Show that Q [x℄=hx2 + 1i is a eld and ompute in this eld the quo-

tient (x3 + x2 + x)=(x3 + x2 + 1), rst by hand and then by using Singular as in Example 1.3.13. Alternatively use the method of Example 1.1.8 (in

hara teristi 0), de ning a minpoly.

1.3.16. Let f = x3 + y3 + z 3 + 3xyz, and let I be the ideal in Q [x; y; z ℄, respe tively F 3 [x; y; z ℄, generated by f and its partial derivatives. Moreover, let R := Q [x; y; z ℄=I and S := F3 [x; y; z ℄=I .

(1) Is xyz a zerodivisor in R, respe tively in S ? (2) Compute the index of nilpoten y of x + y + z in R, respe tively S . (Hint: type ?diff; or ?ja ob; to see how to reate the ideal I .)

1.4 Lo al Rings and Lo alization Lo alization of a ring means enlarging the ring by allowing denominators, similar to the passage from Z to Q . The name, however, omes from the geometri interpretation. For example, lo alizing K [x1 ; : : : ; xn ℄ at hx1 ; : : : ; xn i means onsidering rational fun tions f=g where f and g are polynomials with g(0) 6= 0. Of ourse, any polynomial f = f=1 is of this form but, as g may have zeros arbitrary lose to 0, f=g is de ned only lo ally, in an arbitrary small neighbourhood of 0 ( f. Appendix A.8). 3 This is the usual way to de ne an Artinian ring .

1.4 Lo al Rings and Lo alization

31

De nition 1.4.1. A ring A is alled lo al if it has exa tly one maximal

ideal m. A=m is alled the residue eld of A. Rings with nitely many maximal ideals are alled semi{lo al . We denote lo al rings also by (A; m) or (A; m; K ) where K = A=m.

Fields are lo al rings. A polynomial ring K [x1 ; : : : ; xn ℄ with n 1 over a eld K is, however, never lo al. To see this, onsider for any (a1 ; : : : ; an ) 2 K n the ideal ma := hx1 a1 ; : : : ; xn an i. Sin e ' : K [x1 ; : : : ; xn ℄ ! K [x1 ; : : : ; xn ℄, '(xi ) := xi ai , is an isomorphism sending m0 = hx1 ; : : : ; xn i to ma , it follows that K [x1 ; : : : ; xn ℄=ma = K is a eld, hen e ma is a maximal ideal. Sin e K has at least two elements, K n has at least two dierent points and, hen e, K [x1 ; : : : ; xn ℄ has at least as many maximal ideals as K n points (those of type ma ). If K is algebrai ally losed, then the ideals ma , a 2 K n are all maximal ideals of K [x1 ; : : : ; xn ℄ (this is one form of Hilbert's Nullstellensatz). A typi al lo al ring is the formal power series ring K [[x1 ; : : : ; xn ℄℄ with maximal ideal m = hx1 ; : : : ; xn i, that is, all power series without onstant term. That this ring is lo al follows easily from Lemma 1.4.3. We shall treat power series rings in Chapter 6. Other examples are lo alizations of polynomial rings at prime ideals, f. Example 1.4.6.

Theorem 1.4.2. Every ring A 6= 0 ontains at least one maximal ideal. If I ( A is an ideal, then there exists a maximal ideal

m A su h that I m.

Proof. The rst statement follows from the se ond with I = 0. If I is not maximal there exists an f1 2 A su h that I ( I1 := hI; f1 i ( A. If I1 is not maximal there is an f2 su h that I1 ( I2 = hI1 ; f2 i ( A. Continuing in this manner, we obtain a sequen e of stri tly in reasing ideals I ( I1 ( I2 ( : : : whi h must be ome stationary, say Im = In for m n if A is Noetherian by Proposition 1.3.6. Thus, In is maximal and ontains I . In general, if A is S not Noetherian, n1 In is an ideal ontaining I , and the result follows from Zorn's lemma.4

Lemma 1.4.3. Let A be a ring. (1) A is a lo al ring if and only if the set of non{units is an ideal (whi h is then the maximal ideal). (2) Let m A be a maximal ideal su h that every element of the form 1 + a, a 2 m is a unit. Then A is lo al.

Proof. (1) is obvious. To see (2) let u 2 A r m. Sin e m is maximal hm; ui = A and, hen e, 1 = uv + a for some v 2 A, a 2 m. By assumption uv = 1 a is a unit. Hen e, u is a unit and m is the set of non{units. The laim follows from (1). 4 Zorn's Lemma says: let S be a non{empty system of sets su h that for ea h hain I1 I2 : : : In : : : in S , the union of the hain elements belong to S . Then any element of S is ontained in a maximal element (w.r.t. in lusion) of S . This \lemma" is a tually an axiom, equivalent to the axiom of hoi e.

32


Lo alization generalizes the onstru tion of the quotient eld: if A is an integral domain, then the set Quot(A) := Q(A) :=

a a; b 2 A; b 6= 0 ; b

together with the operations

a a0 ab0 + a0 b a a0 aa0 ; + 0 = = 0 b b bb b b0 bb0 is a eld, the quotient eld or eld of fra tions of A. Here a=b denotes the

lass of (a; b) under the equivalen e relation (a; b) (a0 ; b0 ) :() ab0 = a0 b :

The map A ! Q(A), a 7! a=1 is an inje tive ring homomorphism and we identify A with its image. Sin e a=b = 0 if and only if a = 0, every element a=b 6= 0 has an inverse b=a and, therefore, Q(A) is a eld. The denominators in Q(A) are the elements of the set S = A r f0g and S satis es (1) 1 2 S , (2) a 2 S; b 2 S =) ab 2 S . This notion an be generalized as follows.

De nition 1.4.4. Let A be a ring.

(1) A subset S A is alled multipli ative or multipli atively losed if onditions (1) and (2) above hold. (2) Let S A be multipli atively losed. We de ne the lo alization or the ring of fra tions S 1 A of A with respe t to S as follows:

S 1 A :=

a a 2 A; b 2 S b

where a=b denotes the equivalen e lass of (a; b) 2 A S with respe t to the following equivalen e relation: (a; b) (a0 ; b0 ) :() 9 s 2 S su h that s(ab0

a0 b) = 0 :

Moreover, on S 1 A we de ne an addition and multipli ation by the same formulas as for the quotient eld above. The following proposition is left as an exer ise.

Proposition 1.4.5. (1) The operations + and on S 1 A are well{de ned (independent of the

hosen representatives) and make S 1 A a ring ( ommutative and with 1 = 1=1).


33

(2) The map j : A ! S 1 A, a 7! a=1 is a ring homomorphism satisfying a) j (s) is a unit in S 1 A if s 2 S, b) j (a) = 0 if and only if as = 0 for some s 2 S,

) j is inje tive if and only if S onsists of non{zerodivisors, d) j is bije tive if and only if S onsists of units. (3) S 1 A = 0 if and only if 0 2 S. (4) If S1 S2 are multipli atively losed in A and onsist of non{zerodivisors, then S1 1A S2 1 A. (5) Every ideal in S 1 A is generated by the image of an ideal in A under the map j. Moreover, the prime ideals in S 1A are in one{to{one orresponden e with the prime ideals in A whi h do not meet S. Examples 1.4.6. (1) A r P is multipli atively losed for any prime ideal P A. The lo alization of A with respe t to A r P is denoted by AP and

a a; b 2 A; b 62 P AP = b

is alled the lo alization of A at the prime ideal P . The set

a a 2 P; b 62 P P AP = b

is learly an ideal in AP . Any element a=b 2 AP r P AP satis es a 62 P , hen e, b=a 2 AP and, therefore, a=b is a unit. This shows that AP is a lo al ring with maximal ideal P AP by Lemma 1.4.3. In parti ular, if m A is a maximal ideal then Am is lo al with maximal ideal mAm . (2) For any f 2 A, the set S := ff n j n 0g is multipli atively losed (with f 0 = 1). We use the spe ial notation

Af := S

1A

a a 2 A; n 0 ; = fn

not to be onfused with Ahf i , if hf i A is a prime ideal. (3) The set S of all non{zerodivisors of A is multipli atively losed. For this S , S 1 A =: Q(A) =: Quot(A) is alled the total ring of fra tions or the total quotient ring of A. If A is an integral domain, this is just the quotient eld of A. Two spe ial but important ases are the following: if K [x1 ; : : : ; xn ℄ is the polynomial ring over a eld, then the quotient eld is denoted by K (x1 ; : : : ; xn ),

K (x1 ; : : : ; xn ) := Q(K [x1 ; : : : ; xn ℄) ;

34


whi h is also alled the fun tion eld in n variables; the xi are then also alled parameters . For omputing with parameters f. Singular-Example 1.1.8. The lo alization of K [x℄ = K [x1 ; : : : ; xn ℄ with respe t to the maximal ideal hxi = hx1 ; : : : ; xn i is

f f; g 2 K [x℄; g(0) 6= 0 : K [x℄hxi = g It is an important fa t that we an ompute in this ring without expli it denominators, just by de ning a suitable monomial ordering on K [x℄ ( f. Se tion 1.5). More generally, we an ompute in K [x℄ma , ma = hx1 a1 ; : : : xn an i, for any a = (a1 ; : : : ; an ) 2 K n , by translating our polynomial data to K [x℄hxi via the ring map xi 7! xi + ai .

Proposition 1.4.7. Let ' : A ! B be a ring homomorphism, S A multipli atively losed, and j : A ! S 1 A the anoni al ring homomorphism a 7! a=1.

(1) Assume (i) '(s) is a unit in B for all s 2 S. Then there exists a unique ring homomorphism the following diagram ommutes: A EE

'

EE EE j EE"

: S 1 A ! B su h that

/B y< y yy yy y y

S 1A

(2) Assume moreover (ii) '(a) = 0 implies sa = 0 for some s 2 S, (iii) every element of B is of the form '(a)'(s) 1 . Then is an isomorphism. Property (1) is alled the universal property of lo alization .

Proof. (1) Sin e '(a) = (a=1) for a 2 A, we obtain, for any a=s 2 S 1 A, that (a=s) = (a=1) (1=s) = (a=1) (s=1) 1 = '(a)'(s) 1 . In parti ular, is unique if it exists. Now de ne (a=s) := '(a)'(s) 1 and he k that is well{de ned and a ring homomorphism. (2) (ii) implies that is inje tive and (iii) that

is surje tive.

Lemma 1.4.8. Let S A be multipli atively losed and j : A ! S 1 A the

anoni al ring homomorphism a 7! a=1.

(1) If J S 1 A is an ideal and I = j 1 (J ) then IS 1 A = J. In parti ular, if f1; : : : ; fk generate I over A then f1 ; : : : ; fk generate J over S 1A. (2) If A is Noetherian, then S 1 A is Noetherian.


35

Proof. (1) If f=s 2 J then f=1 = s f=s 2 J , hen e f 2 I = j 1 (J ) and, therefore, f=s = f 1=s 2 IS 1 A. The other in lusion is lear. Statement (2) follows dire tly from (1).

To de ne the lo al ring K [x℄hxi = K [x1 ; : : : ; xn ℄hx1 ;:::;xn i in Singular, we have to hoose a lo al ordering su h as ds, Ds, ls or a weighted lo al ordering. This is explained in detail in the next se tion. We shall now show the dieren e between lo al and global rings by some examples. Note that obje ts de ned in the lo al ring K [x℄hxi ontain geometri information (usually only) about a Zariski neighbourhood of 0 2 K n ( f. A.2, page 414), while obje ts in K [x℄ ontain geometri information whi h is valid in the whole aÆne spa e K n. Consider the ideal I = hy (x 1); z (x 1)i Q [x; y; z ℄ and onsider the

ommon zero{set of all elements of I ,

V (I ) = f(x; y; z ) 2 C 3 j f (x; y; z ) = 0 8 f 2 I g = f(x; y; z ) 2 C 3 j y (x 1) = z (x 1) = 0g : The real pi ture of V (I ) is displayed in Figure 1.1.

0

Fig. 1.1.

1

x

The real zero{set of hy (x 1);z (x 1)i

Although we treat dimension theory later, it should be intuitively lear from the pi ture that the (lo al) dimension of V (I ) is 1 at the point (0; 0; 0) and 2 at the point (1; 0; 0). We ompute the global dimension of V (I ) (whi h is the maximum of the dimensions at ea h point) and then the dimension of V (I ) in the points (0; 0; 0) and (1; 0; 0). As we shall see in Se tion 3.3, we always have to ompute a standard basis of the ideal with respe t to the given ordering rst and then apply the ommand dim.

SINGULAR Example 1.4.9 (global versus lo al rings).

(1) Compute the dimension of V (I ), that is, ompute dim(I ), the Krull dimension of S=I , S = Q [x; y; z ℄ ( f. Chapter 3, Se tion 3.3).

36

1. Rings, Ideals and Standard Bases ring S = 0,(x,y,z),dp; ideal I = y*(x-1), z*(x-1); ideal J = std(I); // ompute a standard basis J of I in S J; //J = //-> J[1℄=xz-z //-> J[2℄=xy-y dim(J); //the (global) dimension of V(I) is 2 //-> 2 redu e(y,J); //-> y

//y is not in I //(result is 0 iff y is in I)

(2) Compute the dimension of V (I ) at 0 = (0; 0; 0), that is, ompute dim(I ), the Krull dimension of R=IR, R = Q [x; y; z ℄hx;y;zi . ring R = 0,(x,y,z),ds; ideal I = fet h(S,I);//fet h I from S to basering ideal J = std(I); // ompute a standard basis J of I in R J; //-> J[1℄=y //J = sin e x-1 is a unit in R //-> J[2℄=z dim(J); //-> 1 //(lo al) dimension of V(I) at 0 is 1 redu e(y,J); //-> 0 //now y is in IR = JR

(3) Compute the dimension of V (I ) at (1; 0; 0), that is, ompute dim(I1 ), the Krull dimension of R=I1 R in R = Q [x; y; z ℄hx;y;zi where I1 is the translation of I to (1; 0; 0). map trans = S, x+1,y,z; //repla e x by x+1 and leave //y,z fixed, i.e., translate //(0,0,0) to (1,0,0) ideal I1 = trans(I); I1; //-> I1[1℄=xy //-> I1[2℄=xz dim(std(I1)); //dimension of V(I) at (1,0,0) is 2 //-> 2

(4) Compute the (global) dimension of V (I ) after translation. setring S; //go ba k to global ring S map trans = S, x+1,y,z; ideal I1 = trans(I); //translate I, as in (3) I1;

1.4 Lo al Rings and Lo alization //-> I1[1℄=xy //-> I1[2℄=xz dim(std(I1)); //-> 2

37

//(global) dimension of translated //variety has not hanged

kill S,R;

The above omputation illustrates what is intuitively lear from the pi ture in Figure 1.1: the dimension of the lo al rings varies. Dimension theory is treated in detail in Chapter 3, Se tion 3.3. For this example, it is enough to have an intuitive feeling for the dimension as it is visualized in the real pi ture of V (I ).

Exer ises 1.4.1. Prove Proposition 1.4.5. 1.4.2. Let A be a ring, I A an ideal and f 2 A.

Prove that IAf \ A is the saturation of I with respe t to f , that is, equal to I : hf i1 = fg 2 A j 9 n su h that gf n 2 I g.

1.4.3. Let (A; m) be a lo al K {algebra, K a eld, and I that dimK (A=I ) < 1. Show that mn I for some n.

A an ideal su h

1.4.4. Let A be a ring and J (A) the interse tion of all maximal ideals of A,

whi h is alled the Ja obson radi al of A. Prove that for all x 2 J (A), 1 + x is a unit in A.

1.4.5. Let S and T be two multipli atively losed sets in the ring A. Show

that ST is multipli atively losed and that (ST ) 1 (A) and T 1 (S 1 A) are isomorphi , if T denotes also the image of T in S 1 A. In parti ular, if S T then T 1 A = T 1 (S 1 A). Hen e, for Q P two prime ideals we obtain AQ = (AP )QAP .

1.4.6. Let S A be the set of non{zerodivisors. Show the following statements about the total ring of fra tions Quot(A) = S 1 A:

(1) S is the biggest multipli atively losed subset of A su h that A ! S 1 A is inje tive. (2) Ea h element of Quot(A) is either a unit or a zerodivisor. (3) A ring A, su h that ea h non{unit is a zerodivisor, is equal to its total ring of fra tions, that is, A ! Quot(A) is bije tive.

1.4.7. (1) Consider the two rings A = C [x; y℄=hx2 and the multipli ative sets:

y3 i and B = C [x; y℄=hxyi

38


S the set of non{zerodivisors of A, respe tively of B , and T := A r hx; yiA , respe tively T := B r hx; yiB . Determine the lo alizations of A and B with respe t to T and S . (2) Are any two of the six rings A; B , S 1 A, S 1 B , T 1 A, T 1 B isomorphi ? 1.4.8. Let A be a ring and B = A=(PL 1 \ \ Pr ) with Pi A prime ideals. Show that the rings Quot(B ) and ri=1 Quot(A=Pi ) are isomorphi . In parti ular, Quot(B ) is a dire t sum of elds. (Hint: use Exer ise 1.3.13.)

1.4.9. Let A be a unique fa torization domain (that is, A is a domain and every non{unit of A an be written as a produ t of irredu ible elements su h that the fa tors are uniquely determined up to multipli ation with units) and S A multipli atively losed. Show that S 1 A is a unique fa torization domain. (Hint: enlarge, if ne essary, S to a multipli ative system Se su h that (1) Se 1 A = S 1 A and (2) if s 2 Se and s = s1 s2 then s1 ; s2 2 Se.) 1.4.10. Let A be an integral domain. Then, for any prime ideal P A, we

onsider the lo alization AP as a subring of the quotient eld Quot(A) and, hen e, we an onsider their interse tion. Show that

A=

\

P 2Spe A

AP =

\

m2Max A

Am :

1.4.11. Let I = I1 I2 I3 Q[x; y; z ℄ be the produ t of the ideals I1 = hz x2 i,

I2 = hy; z i and I3 = hxi. Compute, as in Singular Example 1.4.9, the dimension of V (I ) at the points (0; 0; 0), (0; 0; 1), (1; 0; 0) and (1; 1; 1). Draw a real pi ture of V (I ) and interpret your results geometri ally.

1.5 Rings Asso iated to Monomial Orderings In this se tion we show that non{global monomial orderings lead to new rings whi h are lo alizations of the polynomial ring. This fa t has far{rea hing

omputational onsequen es. For example, hoosing a lo al ordering, we an, basi ally, do the same al ulations in the lo alization of a polynomial ring as with a global ordering in the polynomial ring itself. In parti ular, we an ee tively ompute in K [x1 ; : : : ; xn ℄hx1 ;:::;xk i for k n (by Lemma 1.5.2 (3) and Example 1.5.3). Let > be a monomial ordering on the set of monomials Mon(x1 ; : : : ; xn ) = fx j 2 N n g, and K [x℄ = K [x1 ; : : : ; xn ℄ the polynomial ring in n variables over a eld K . Then the leading monomial fun tion LM has the following properties for polynomials f; g 2 K [x℄ n f0g:

1.5 Rings Asso iated to Monomial Orderings

39

(1) LM(gf ) = LM(g ) LM(f ). (2) LM(g + f ) maxfLM(g ); LM(f )g with equality if and only if the leading terms of f and g do not an el. In parti ular, it follows that

S> := fu 2 K [x℄ r f0g j LM(u) = 1g is a multipli atively losed set.

De nition 1.5.1. For any monomial ordering > on Mon(x1 ; : : : ; xn ), we de ne

K [x℄> := S>

1 K [x℄

f f; u 2 K [x℄; LM(u) = 1 ; = u

the lo alization of K [x℄ with respe t to S> and all K [x℄> the ring asso iated

to K [x℄ and >.

Note that S> = K if and only if > is global and S> = K [x℄ r hx1 ; : : : ; xn i if and only if > is lo al.

Lemma 1.5.2. Let K be a eld, K [x℄ = K [x1 ; : : : ; xn ℄, and let > be a monomial ordering on Mon(x1 ; : : : ; xn ). Then

(1) K [x℄ K [x℄> K [x℄hxi . (2) The set of units in K [x℄> is given by

v (K [x℄> ) = j u; v 2 K [x℄; LM(v) = LM(u) = 1 ; u and satis es (K [x℄> ) \ K [x℄ = S> . (3) K [x℄ = K [x℄> if and only if > is a global ordering and K [x℄> = K [x℄hxi if and only if > is a lo al ordering. (4) K [x℄> is a Noetherian ring. (5) K [x℄> is fa torial. We shall see later (Corollary 7.4.6) that the in lusions of Lemma 1.5.2 (1) are at ring morphisms.

Proof. (1) The rst in lusion is lear by Proposition 1.4.5 (2), the se ond follows from Proposition 1.4.5 (4) sin e LM(u) = 1 implies u 62 hxi. (2) If f=u is a unit in K [x℄> , there is a h=v su h that (f=u) (h=v ) = 1. Hen e, fh = uv and LM(f ) LM(h) = 1, whi h implies LM(f ) = 1. (3) By Proposition 1.4.5 (2), K [x℄ = K [x℄> if and only if S> onsists of units of K [x℄, that is if and only if S> K , whi h is equivalent to > being global. The se ond equality follows sin e K [x℄ r hxi onsists of units in K [x℄> if and only if every polynomial with non{zero onstant term belongs to S> whi h is equivalent to > being lo al.

40


(4) Follows from Lemma 1.4.8. (5) Sin e K [x℄ is fa torial, this follows from Exer ise 1.4.9.

Examples 1.5.3. We des ribe some familiar and some less familiar rings, asso iated to a polynomial ring and a monomial ordering.

(1) Let K [x; y ℄ = K [x1 ; : : : ; xn ; y1 ; : : : ; ym ℄ and onsider the produ t ordering > = (>1 ; >2 ) on Mon(x1 ; : : : ; xn ; y1 ; : : : ; ym ), where >1 is global on Mon(x1 ; : : : ; xn ) and >2 is lo al on Mon(y1 ; : : : ; ym ). Then

x y > 1 > y for all ; 6= 0; all and hen e S> = K + hy i K [y ℄. It follows that K [x; y℄> = (K [y℄hyi)[x℄ ;

whi h equals K [y ℄hyi K K [x℄ ( f. Se tion 2.7 for the tensor produ t). (2) Now let >1 be lo al and >2 global, > = (>1 ; >2 ), then

x y < 1 < y for all ; 6= 0; all and hen e S> = K + hxiK [x; y ℄. We obtain stri t in lusions (K [x℄hxi )[y ℄ ( K [x; y ℄> ( K [x; y ℄hxi ; sin e 1=(1 + xy ) is in the se ond but not in the rst and 1=y is in the third but not in the se ond ring. (3) If >1 is global, >2 arbitrary and > = (>1 ; >2 ) then S> onsists of elements u 2 K [y ℄ satisfying LM>2 (u) = 1. Hen e,

K [x; y℄> = (K [y℄>2 )[x℄ ( f. Exer ise 2.7.12). This ordering has the following elimination property for x1 ; : : : ; xn :

f

2 K [x; y℄; LM(f ) 2 K [y℄ ) f 2 K [y℄ :

(4) Let > be a lo al ordering on Mon(x1 ; : : : ; xn ) and K (y ) the quotient eld of K [y ℄ = K [y1 ; : : : ; ym ℄. It is not diÆ ult to see that

K (y)[x℄> = K [x; y℄hxi (Exer ise 1.5.6). Hen e, we an ee tively ompute in the lo alization

K [x1 ; : : : ; xn ℄P , where P is a prime ideal generated by a subset of the variables.

De nition 1.5.4. A monomial ordering > on K [x1 ; : : : ; xn ℄ having the elimination property for x1 ; : : : ; xs ( f. Example 1.5.3 (3)) is alled an elimination ordering for x1 ; : : : ; xs .


41

An elimination ordering need not be a produ t ordering but must satisfy xi > 1 for i = 1; : : : ; s (sin e, if xi < 1 then LM(1 + xi ) = 1 but 1 + xi 62 K [xs+1 ; : : : ; xn ℄), that is, an elimination ordering for x1 ; : : : ; xs must be global on Mon(x1 ; : : : ; xs ). Sin e the lexi ographi al ordering is the produ t of the degree orderings on Mon(xi ) for i = 1; : : : ; n, it is an elimination ordering for x1 ; : : : ; xj , for j = 1; : : : ; n. (Cf. Se tion 1.8.2 for appli ations of elimination orderings.) We now extend the leading data to K [x℄> .

De nition 1.5.5. Let > be any monomial ordering: (1) For f 2 K [x℄> hoose u 2 K [x℄ su h that LT(u) = 1 and uf de ne

2 K [x℄. We

LM(f ) := LM(uf ); LC(f ) := LC(uf ); LT(f ) := LT(uf ); LE(f ) := LE(uf ); and tail(f ) = f LT(f ). (2) For any subset G K [x℄> de ne the ideal

L>(G) := L(G) := hLM(g) j g 2 G r f0giK [x℄ : L(G) K [x℄ is alled the leading ideal of G. Remark 1.5.6. (1) The de nitions in 1.5.5 (1) are independent of the hoi e of u. (2) Sin e K [x℄> K [x℄hxi K [[x℄℄, where K [[x℄℄ denotes the formal power series ring ( f. Se tion 6.1), we may onsider f 2 K [x℄> as a formal power series. It follows easily that LM(f ), respe tively LT(f ), orrespond to a unique monomial, respe tively term, in the power series expansion of f . Hen e tail(f ) is the power series of f with the leading term deleted. (3) Note that if I is an ideal, then L(I ) is the ideal generated by all leading monomials of all elements of I and not only by the leading monomials of a given set of generators of I .

Example 1.5.7. (1) Consider Q [x℄ with a lo al ordering (in one variable all lo al, respe tively global, orderings oin ide). For f = 3x=(1 + x) + x we have LM(f ) = x, LC(f ) = 4, LT(f ) = 4x, LE(f ) = 1 and tail(f ) = 3x2 =(1 + x). (2) Let G = ff; g g with f = xy 2 + xy , g = x2 y + x2 y 2 Q [x; y ℄ and monomial ordering dp. If I = hf; g i then L(G) ( L(I ), sin e L(G) = hxy 2 ; x2 y i, but xf yg = y 2 . Thus, y 2 2 L(I ), but y 2 62 L(G).

42


Ring maps between rings asso iated to a monomial ordering are almost as easy as ring maps between polynomial rings.

Lemma 1.5.8. Let : K ! L be a morphism of elds and >1 , >2, monomial orderings on Mon(x1 ; : : : ; xn ) and on Mon(y1 ; : : : ; ym ). Let f1; : : : ; fn 2 L[y1 ; : : : ; ym ℄>2 and assume that, for all h 2 S>1 , h(f1 ; : : : ; fn) 2 S>2 . Then there exists a unique ring map ' : K [x1 ; : : : ; xn ℄>1

! L[y1 ; : : : ; ym ℄>2

satisfying '(xi ) = fi for i = 1; : : : ; n, and '(a) = (a) for a 2 K. Proof. By Lemma 1.1.6, there is a unique ring map 'e : K [x℄ ! L[y℄>2 with 'e(xi ) = fi and 'e(a) = (a), a 2 K . The assumption says that 'e(u) is a unit in L[y ℄>2 for ea h u 2 S>1 . Hen e, the result follows from the universal property of lo alization (Proposition 1.4.7).

In parti ular, if >1 is global, there is no ondition on the fi and any elements f1 ; : : : ; fn 2 K [y1 ; : : : ; ym ℄> de ne a unique map

K [x1 ; : : : ; xn ℄

! K [y1; : : : ; ym ℄> ;

xi 7

! fi ;

for any monomial ordering > on Mon(y1 ; : : : ; ym ).

Remark 1.5.9. With the notations of Lemma 1.5.8, the ondition h 2 S>1 implies h(f1 ; : : : ; fn ) 2 S>2 annot be repla ed by 1 >2 LM(fi ) for those i where 1 >1 xi . Consider the following example: let >1 , respe tively >2 , be de ned on K [x; y ℄ by the matrix 21 10 , respe tively 11 20 . Then x y(1)^10; //the monomial 1 is greater than y(1)^10 //-> 1 ring A2 = 0,(x(1..n),y(1..m)),(ds(n),dp(m)); fet h(A1,f);


43

//-> y(1)^10+y(3)+1+x(1)*y(2)^5+x(1)*x(2)^2 x(1)*y(2)^5 1 ring A3 = 0,(x(1..n),y(1..m)),(dp(n),ds(2),dp(m-2)); fet h(A1,f); //-> x(1)*x(2)^2+x(1)*y(2)^5+y(3)+1+y(1)^10

Exer ises 1.5.1. Prove Remark 1.5.6. 1.5.2. Give one possible realization of the following rings within Singular: (1) Q [x; y; z ℄, (2) F 5 [x; y; z ℄, (3) Q [x; y; z ℄=hx5 + y 3 + z 2 i, (4) Q (i)[x; y ℄, i2 = 1, (5) F 27 [x1 ; : : : ; x10 ℄hx1 ;:::;x10 i , (6) F 32003 [x; y; z ℄hx;y;zi=hx5 + y 3 + z 2 ; xy i, (7) Q (t)[x; y; z ℄, (8) Q [t℄=(t3 + t2 + 1) [x; y; z ℄hx;y;zi, (9) (Q [t℄hti )[x; y; z ℄, (10) F 2 (a; b; )[x; y; z ℄hx;y;zi. (Hint: see the some ideal.)

Singular

Manual for how to de ne a quotient ring modulo

1.5.3. What are the units in the rings of Exer ise 1.5.2? 1.5.4. Write a Singular pro edure, having as input a polynomial f and

returning 1 if f is a unit in the basering and 0 otherwise. (Hint: type ?pro edures;.) Test the pro edure by reating, for ea h ring of Exer ise 1.5.2, two polynomials, one a unit, the other not.

1.5.5. Write a Singular pro edure, having as input a polynomial f and an

integer n, whi h returns the power series expansion of the inverse of f up to terms of degree n if f is a unit in the basering and 0 if f is not a unit. (Hint: remember the geometri series.)

1.5.6. (1) Let > be a lo al ordering on Mon(x1 ; : : : ; xn ). Show that

K [x1 ; : : : ; xn ; y1 ; : : : ; ym℄hx1 ;:::;xni = K (y1 ; : : : ; ym )[x1 ; : : : ; xn ℄> . (2) Realize the ring Q [x; y; z ℄hx;yi inside Singular .

44


1.6 Normal Forms and Standard Bases In this se tion we de ne standard bases, respe tively Grobner bases, of an ideal I K [x℄> as a set of polynomials of I su h that their leading monomials generate the leading ideal L(I ). The next se tion gives an algorithm to

ompute standard bases. For global orderings this is Bu hberger's algorithm, whi h is a generalization of the Gaussian elimination algorithm and the Eu lidean algorithm. For lo al orderings it is Mora's tangent one algorithm, whi h itself is a variant of Bu hberger's algorithm. The general ase is a variation of Mora's algorithm, whi h is due to the authors and implemented in Singular sin e 1990. The leading ideal L(I ) ontains a lot of information about the ideal I , whi h often an be omputed purely ombinatorially from L(I ), be ause the leading ideal is generated by monomials. Standard bases have turned out to be the fundamental tool for omputations with ideals and modules. The idea of standard bases is already ontained in the work of Gordan [75℄. Later, monomial orderings were used by Ma aulay [123℄ and Grobner [94℄ to study Hilbert fun tions of graded ideals, and, more generally, to nd bases of zero{ dimensional fa tor rings. The notion of a standard basis was introdu ed later, independently, by Hironaka [100℄, Grauert [79℄ (for spe ial lo al orderings) and Bu hberger [28℄ (for global orderings). In the following, spe ial emphasis is made to axiomati ally hara terize normal forms, respe tively weak normal forms, whi h play an important role in the standard basis algorithm. They generalize division with remainder to the ase of ideals, respe tively nite sets of polynomials. In the ase of a global ordering, for any polynomial f and any ideal I , there is a unique normal form NF(f j I ) of f with respe t to I , su h that no monomial of NF(f j I ) is in the leading ideal L(I ). This an be used to de ide, for instan e, whether f is in the ideal I (if the normal form is 0). In the general ase, the above property turns out to be too strong. Hen e, the requirements for a normal form have to be weakened. For instan e, for the de ision whether a polynomial is in an ideal or not, only the leading term of a normal form NF(f j I ) is important. Thus, for this purpose, it is enough to require that NF(f j I ) is either 0 or has a leading term, whi h is not in L(I ). After weakening the requirements, there is no more uniqueness statement for the normal form. Our intention is to keep the de nition of a normal form as general as possible. Moreover, our presentation separates the normal form algorithm from a general standard basis algorithm and shows that dierent versions of standard basis algorithms are due to dierent normal forms. The axiomati de nition of a normal form as presented in this se tion has been introdu ed in [90, 91℄, although its properties have already ommonly been used before. It seems to be the minimal requirement in order to arry through standard basis theory in the present ontext.

1.6 Normal Forms and Standard Bases

45

Let > be a xed monomial ordering and let, in this se tion,

R = K [x1 ; : : : ; xn ℄> be the lo alization of K [x℄ = K [x1 ; : : : ; xn ℄ with respe t to >. Re all that R = S> 1K [x℄ with S> = fu 2 K [x℄ r f0g j LM(u) = 1g, and that R = K [x℄ if > is global and R = K [x℄hxi if > is lo al. In any ase, R may be onsidered as a subring of the ring K [[x℄℄ of formal power series ( f. Se tion 6.1).

De nition 1.6.1. Let I R be an ideal.

(1) A nite set G R is alled a standard basis of I if

G I; and L(I ) = L(G) : That is, G is a standard basis, if the leading monomials of the elements of G generate the leading ideal of I , or, in other words, if for any f 2 I r f0g there exists a g 2 G satisfying LM(g ) j LM(f ). (2) If > is global, a standard basis is also alled a Grobner basis . (3) If we just say that G is a standard basis, we mean that G is a standard basis of the ideal hGiR generated by G. Let > be any monomial ordering on Mon(x1 ; : : : ; xn ). Then ea h non{zero ideal I = K [x℄> has a standard basis. To see this, hoose a nite set of generators m1 ; : : : ; ms of L(I ) K [x℄, whi h exists, sin e K [x℄ is Noetherian (Theorem 1.3.5). By de nition of the leading ideal, these generators are leading monomials of suitable elements g1 ; : : : ; gs 2 I . By onstru tion, the set fg1 ; : : : ; gs g is a standard basis for I .

De nition 1.6.2. Let G R be any subset.

(1) G is alled interredu ed if 0 62 G and if LM(g ) - LM(f ) for any two elements f 6= g in G. An interredu ed standard basis is also alled minimal . (2) f 2 R is alled ( ompletely) redu ed with respe t to G if no monomial of the power series expansion of f is ontained in L(G). (3) G is alled ( ompletely) redu ed if G is interredu ed and if, for any g 2 G, LC(g ) = 1 and tail(g ) is ompletely redu ed with respe t to G.

Remark 1.6.3. (1) If > is a global ordering, then any nite set G an be transformed into an interredu ed set: for any g 2 G su h that there exists an f 2 G rfg g with LM(f ) j LM(g ) repla e g by g mf , where m is a term with LT(g ) = m LT(f ). The result is alled the (inter)redu tion of G; it generates the same ideal as G.

46


(2) Every standard basis G an be transformed into an interredu ed one by just deleting elements of G: delete zeros and then, su

essively, any g su h that LM(g ) is divisible by LM(f ) for some f 2 G r fg g. The result is again a standard basis. Thus, G is interredu ed if and only if G is minimal (that is, we annot delete any element of G without violating the property of being a standard basis). (3) Let G R be an interredu ed set and g 2 G. If tail(g ) is not redu ed with respe t to G, the power series expansion of tail(g ) has a monomial whi h is either divisible by L(g ) or by L(f ) for some f 2 G rfg g. If > is global, then no monomial of tail(g ) is divisible by L(g ) sin e > re nes the natural partial ordering on N n , that is, tail(g ) is redu ed with respe t to fgg. For lo al or mixed orderings, however, it is possible to redu e tail(g) with g and we a tually have to do this. (4) It follows that a Grobner basis G K [x℄, whi h onsists of moni polynomials, is ( ompletely) redu ed if for any f 6= g 2 G, LM(g ) does not divide any monomial of f . We shall see later that redu ed Grobner bases an always be omputed5 ( f. the remark after Algorithm 1.6.10), and are unique (Exer ise 1.6.1), but redu ed standard bases are, in general, not omputable (in a nite number of steps). The following two de nitions are ru ial for our treatment of standard bases.

De nition 1.6.4. Let G denote the set of all nite lists G R. A map NF : R G ! R; (f; G) 7! NF(f j G) ; is alled a normal form on R if, for all G 2 G ,

(0) NF(0 j G) = 0 ,

2 R and G 2 G , NF(f j G) = 6 0 =) LM NF(f j G) 62 L(G). If G = fg1 ; : : : ; gs g, then r := f NF(f j G) has a standard representa-

and, for all f (1) (2)

tion with respe t to G, that is, either r = 0, or r=

s X i=1

ai gi ; ai 2 R ;

satisfying LM(f ) LM(ai gi ) for all i su h that ai gi 6= 0. NF is alled a redu ed normal form , if, moreover, NF(f j G) is redu ed with respe t to G.

5 The Singular ommand std an be for ed to ompute redu ed Grobner bases G (up to normalization) using option(redSB). To normalize G, one may use the Singular ommand simplify(G,1).


47

De nition 1.6.5.

(1) A map NF : R G ! R, as in De nition 1.6.4, is alled a weak normal form on R if it satis es (0),(1) of 1.6.4 and, instead of (2), (2') for all f 2 R and G 2 G there exists a unit u 2 R su h that

uf

NF(f j G)

has a standard representation with respe t to G. (2) A weak normal form NF is alled polynomial if, whenever f 2 K [x℄ and G K [x℄, there exists a unit u 2 R \ K [x℄ su h that uf NF(f j G) has a standard representation with ai 2 K [x℄.

Remark 1.6.6. (1) The notion of weak normal forms is only interesting for non{global orderings sin e for global orderings we have R = K [x℄ and, hen e, R = K . Even in general, if a weak normal form NF exists, then, theoreti ally, f there exists also a normal form NF 1 f f j G) (f; G) 7 ! NF(f j G) =: NF(

u for an appropriate hoi e of u 2 R (depending on f and G). However,

(2)

(3) (4) (5)

we are really interested in polynomial normal forms, and for non{global orderings 1=u is, in general, not a polynomial but a power series. Note that R \ K [x℄ = S> . Consider f = y, g = (y x)(1 y ), and G = fg g in R = K [x; y ℄hx;yi with lo al ordering ls. Assume h := NF(f j G) 2 K [x; y ℄ is a polynomial normal form of f with respe t to G. Sin e f 62 hGiR = hy xiR , we have h 6= 0, hen e, LM(h) 62 L(G) = hyi. Moreover h y = h f 2 hy xiR , whi h implies LM(h) < 1. Therefore, we obtain h = xh0 for some h0 (be ause of the hosen ordering ls). However, y xh0 62 h(y x)(1 y )iK [x;y℄ (substitute (0; 1) for (x; y )) and, therefore no polynomial normal form of (f; G) exists. On the other hand, setting u = (1 y ) and h = x(1 y ) then uy h = (y x)(1 y ) and, hen e, h is a polynomial weak normal form. For appli ations (weak) normal forms are most useful if G is a standard basis of hGiR . We shall demonstrate this with a rst appli ation in Lemma 1.6.7. P f = i ai gi being a standard representation means that no an ellation of leading terms > LM(f ) between the ai gi an o

ur and that LM(f ) = LM(ai gi ) for at least one i. We do not distinguish stri tly between lists and (ordered) sets. Sin e, in the de nition of normal form, we allow repetitions of elements in G we need lists, that is, sequen es of elements, instead of sets. We assume a given set G to be ordered (somehow) when we apply NF( j G).

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Lemma 1.6.7. Let I R be an ideal, G I a standard basis of I and

j G) a weak normal form on R with respe t to G. (1) For any f 2 R we have f 2 I if and only if NF(f j G) = 0. (2) If J R is an ideal with I J, then L(I ) = L(J ) implies I = J. (3) I = hGiR , that is, the standard basis G generates I as R{ideal. (4) If NF( j G) is a redu ed normal form, then it is unique. 6 Proof. (1) If NF(f j G) = 0 then uf 2 I and, hen e, f 2 I . If NF(f j G) 6= 0, then LM NF(f j G) 62 L(G) = L(I ), hen e NF(f j G) 62 I , whi h implies f 62 I , sin e hGiR I . (2) Let f 2 J and assume NF(f j G) = 6 0. Then LM NF(f j G) 62 L(G) = L(I ) = L(J ), a ontradi tion, sin e NF(f j G) 2 J . Hen e, f 2 I by (1). (3) Sin e L(I ) = L(G) L(hGiR ) L(I ), G is also a standard basis of hGiR , NF(

and the statement follows from (2). (4) Let f 2 R and assume that h; h0 are two redu ed normal forms of f with respe t to G. Then no monomial of the power series expansion of h or h0 is divisible by any monomial of L(G) and, moreover,

h h0 = (f

h0 ) (f

h) 2 hGiR = I :

If h h0 6= 0, then LM(h h0 ) 2 L(I ) = L(G), a ontradi tion, sin e LM(h h0 ) is a monomial of either h or h0 .

Remark 1.6.8. The above properties are well{known for Grobner bases with R = K [x℄. For lo al or mixed orderings it is quite important to work onsequently with R instead of K [x℄. We give an example showing that none of the above properties (1) { (3) holds for K [x℄, if they make sense, that is, if the input data are polynomial. Let f1 := x10 x9 y 2 , f2 := y 8 x2 y 7 , f3 := x10 y 7 , and onsider the (lo al) ordering ds on K [x; y ℄. Then R = K [x; y ℄hx;yi , (1 xy )f3 = y 7 f1 + x9 yf2 , and we set

I := hf1 ; f2 iR = hf1 ; f2 ; f3 iR ; I 0 := hf1 ; f2 iK [x;y℄ ; J 0 := hf1 ; f2 ; f3iK [x;y℄; and G := ff1 ; f2 g. Then G is a redu ed standard basis of I (sin e we must multiply f1 at least with y 8 and f2 with x10 to produ e new monomials, but L(G) hx; y i17 ). If NF( j G) is any weak normal form on R, then NF(f3 j G) = 0, sin e f3 2 I . Hen e, we have in this ase (1) NF(f3 j G) = 0, but f3 62 I 0 , (2) I 0 J 0 , L(I 0 ) = L(J 0 ), but I 0 6= J 0 . (3) G J 0 , but hGiK [x℄ 6= J 0 , 6 In the ase of a global ordering, we shall see below that su h a redu ed normal form exists. Then we also write NF( j I ) for NF( of I , and all it the normal form with respe t to I .

j G), G any standard basis


49

Note that J 0 is even hx; y i{primary (for a de nition f. Chapter 4). We on entrate rst on well{orderings, Grobner bases and Bu hberger's algorithm . To des ribe Bu hberger's normal form algorithm, we need the notion of an s{polynomial, due to Bu hberger.

De nition 1.6.9. Let f; g 2 R r f0g with LM(f ) = x and LM(g) = x , respe tively. Set

:= l m(; ) := max(1 ; 1 ); : : : ; max(n ; n )

and let l m(x ; x ) := x be the least ommon multiple of x and x . We de ne the s{polynomial (spoly , for short) of f and g to be spoly(f; g ) := x f

LC(f ) x g: LC(g )

If LM(g ) divides LM(f ), say LM(g ) = x , LM(f ) = x , then the s{polynomial is parti ularly simple, spoly(f; g ) = f

LC(f ) x g; LC(g )

and LM spoly(f; g ) < LM(f ). For the normal form algorithm, the s{polynomial will only be used in the se ond form, while for the standard basis algorithm we need it in the general form above. In order to be able to use the same expression in both algorithms, we prefer the above de nition of the s{polynomial and not the symmetri form LC(g )x f LC(f )x g . Both are, of ourse, equivalent, sin e we work over a eld K . However, in onne tion with pseudo standard bases (Exer ise 2.3.6) we have to use the symmetri form.

Algorithm 1.6.10 (NFBu hberger(f j G)).

Assume that > is a global monomial ordering. Input:

Output:

f

2 K [x℄; G 2 G

h 2 K [x℄, a normal form of f with respe t to G.

h := f ; while (h 6= 0 and Gh := fg 2 G j LM(g ) divides LM(h)g 6= ;)

hoose any g 2 Gh ; h := spoly(h; g); return h;

Note that ea h spe i hoi e of \any" an give a dierent normal form fun tion.

50


Proof. The algorithm terminates, sin e in the i{th step of the while loop we

reate (setting h0 := f ) an s{polynomial hi = h i

1

mi gi ; LM(hi 1 ) > LM(hi ) ;

where mi is a term su h that LT(mi gi ) = LT(hi 1 ), and gi 2 G (allowing repetitions). Sin e > is a well{ordering, fLM(hi )g has a minimum, whi h is rea hed at some step m. We obtain

h1 = f m1 g1 h2 = h1 m2 g2 = f .. .

hm = f

m X i=1

m1 g1

m2 g2

mi gi ;

satisfying LM(f ) = LM(m1 g1 ) > LM(mi gi ) > LM(hm ). This shows that h := hm is a normal form with respe t to G. Moreover, if h 6= 0, then Gh = ; and, hen e, LM(h) 62 L(G) if h 6= 0. This proves orre tness, independent of the spe i hoi e of \any" in the while loop. It is easy to extend NFBu hberger to a redu ed normal form. Either we do tail{redu tion during NFBu hberger, that is, we set

h = spoly(h; g); h = LT(h) + NFBu hberger(tail h) j G ; in the while loop, or do tail{redu tion after applying NFBu hberger, as in Algorithm 1.6.11. Indeed, the argument holds for any normal form with respe t to a global ordering.

Algorithm 1.6.11 (redNFBu hberger(f j G)). Assume that > is a global monomial ordering. Input: Output:

f 2 K [x℄, G 2 G h 2 K [x℄, a redu ed normal form of f with respe t to G

h := 0; g := f ; while (g 6= 0) g := NFBu hberger (g j G); if (g 6= 0) h := h + LT(g); g := tail(g); return h= LC(h);


51

Sin e tail(g ) has stri tly smaller leading term than g , the algorithm terminates, sin e > is a well{ordering. Corre tness follows from the orre tness of NFBu hberger.

Example 1.6.12. Let > be the ordering dp on Mon(x; y; z ), f = x3 + y2 + 2z 2 + x + y + 1 ; G = fx; yg : NFBu hberger

pro eeds as follows:

LM(f ) = x3 , Gf = fxg, h1 = spoly(f; x) = y2 + 2z 2 + x + y + 1, LM(h1 ) = y 2 , Gh1 = fy g, h2 = spoly(h1 ; y) = 2z 2 + x + y + 1, Gh2 = ;.

Hen e, NFBu hberger(f j G) = 2z 2 + x + y + 1. For the redu ed normal form in Algorithm 1.6.11 we obtain:

g0 = NFBu hberger(f j G) = 2z 2 + x + y + 1, LT(g0 ) = 2z 2, h1 = 2z 2; g1 = tail(g0 ) = x + y + 1, g2 = NFBu hberger(g1 j G) = 1, LT(g2 ) = 1, h2 = 2z 2 + 1, g3 = tail(g2 ) = 0. Hen e, RedNFBu hberger(f j G) = z 2 + 1=2.

SINGULAR Example 1.6.13 (normal form).

Note that NF(f j G) may depend on the sorting of the elements of G. The fun tion redu e omputes a normal form. ring A = 0,(x,y,z),dp; //a global ordering poly f = x2yz+xy2z+y2z+z3+xy; poly f1 = xy+y2-1; poly f2 = xy; ideal G = f1,f2; ideal S = std(G); //a standard basis of S; //-> S[1℄=x //-> S[2℄=y2-1 redu e(f,G); //** G is no standardbasis //-> y2z+z3 //NF w.r.t. a non-standard basis G=f2,f1; redu e(f,G); //** G is no standardbasis //-> y2z+z3-y2+xz+1 //NF for a different numbering in G

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1. Rings, Ideals and Standard Bases redu e(f,S,1); //-> z3+xy+z

//NFBu hberger

redu e(f,S); //-> z3+z

//redNFBu hberger

Remark 1.6.14. There exists also the notion of a standard basis over a ring . Namely, let R be Noetherian and R[x℄ = R[x1 ; : : : ; xn ℄. The leading data of f 2 R[x1 ; : : : ; xn ℄ r f0g with respe t to a monomial ordering > on Mon(x1 ; : : : ; xn ) are de ned as in De nition 1.2.2. If I R[x℄ is an ideal and G I a nite set, then G is a standard basis of I if

hLT(f ) j f 2 I i = hLT(g) j g 2 Gi : Note that we used leading terms and not leading monomials (whi h is, of

ourse, equivalent if R is a eld). The normal form algorithm over rings is more ompli ated than over elds. For example, if > is a global ordering, the algorithm NFBu hberger has to be modi ed to h := f ; while (h 6= 0 and Gh = fg1 ; : : : ; gs g 6= ; and LT(h) 2 hLT(g ) j g 2 Gh i)

hoose i 2 R r f0g and monomials mi with mi LM(gi ) = LM(h) su h that Ps LT(h) = 1 m1 LT(g1 ) + + s ms LT(gs ); h := h i=1 i mi gi ; return h; The determination of the i requires the solving of linear equations over R and not just a divisibility test for monomials as for s{polynomials. With this normal form, standard bases an be omputed as in the next se tion. For details see [1℄, [72℄, [105℄. In pra ti e, however, this notion is not frequently used so far and there seems to be no publi ly available system having this implemented. A weaker

on ept are the omprehensive Grobner bases of Weispfenning [183℄, whi h are Grobner bases depending on parameters and whi h spe ialize to a Grobner basis for all possible xed values of the parameters. For a simple riterion, when the spe ialization of a standard basis is again a standard basis, see Exer ises 2.3.7, 2.3.8, where we introdu e pseudo standard bases .

Exer ises Let > be any monomial ordering and R = K [x1 ; : : : ; xn ℄> .

1.6.1. Let I R be an ideal. Show that if I has a redu ed standard basis, then it is unique.

1.7 The Standard Basis Algorithm

53

1.6.2. Let > be a lo al or mixed ordering. Prove that Algorithm 1.6.11 omputes, theoreti ally, (possibly in in nitely many steps) for f 2 R and G R a redu ed normal form. Hen e, it an be used to ompute, for lo al degree orderings, a normal form whi h is ompletely redu ed up to a nite, but arbitrarily high order. 1.6.3. Show by an example, with f and G onsisting of polynomials and > not global, that a ompletely redu ed normal form of f with respe t to G does not exist in R. (Note that Exer ise 1.6.2 only says that it exists as

formal power series.)

1.6.4. Apply NFBu hberger to (f; G; >) without using Singular:

(1) f = 1, G = fx 1g and ordering lp, respe tively ls. (2) f = x4 + y 4 + z 4 + xyz , G = ff=x; f=y; f=z g and ordering dp.

1.6.5. Give a dire t argument that the set G in Exer ise 1.6.4 (2) is a standard basis with respe t to dp.

1.6.6. Write a Singular pro edure, having two polynomials f; g as input and returning spoly(f; g ) as output.

1.6.7. Write your own Singular pro edure, having a polynomial f and an

ideal I as input and NFBu hberger (f j I ) as output by always hoosing the rst element from Gh . (Note that an ideal is given by a list of polynomials.)

1.6.8. Implement, as Singular pro edures, the two ways des ribed in the text to ompute a redu ed normal form. (The rst method is a good exer ise in re ursive programming.) Che k your pro edures with the Singular Example 1.6.13. 1.6.9. Let R = K [t1 ; : : : ; tn ℄, K a eld. Write a Singular pro edure whi h

omputes the normal form NFBu hberger over the ring R, as explained in Remark 1.6.14. (Hint: use the Singular ommand lift.)

1.7 The Standard Basis Algorithm Let > be a xed monomial ordering and let, in this se tion,

R = K [x1 ; : : : ; xn ℄> be the lo alization of K [x℄, x = (x1 ; : : : ; xn ), with respe t to >. Re all that R = S> 1 K [x℄ with S> = fu 2 K [x℄ r f0g j LM(u) = 1g, and that R = K [x℄ if > is global and R = K [x℄hxi if > is lo al. In any ase, R may be onsidered as a subring of the ring K [[x℄℄ of formal power series.

The idea of many standard basis algorithms may be formalized as follows:

54


Algorithm 1.7.1 (Standard(G,NF)).

Let > be any monomial ordering, and R := K [x1 ; : : : ; xn ℄> . Input: Output:

G 2 G , NF an algorithm returning a weak normal form. S 2 G su h that S is a standard basis of I = hGiR R

S = G; P = f(f; g) j f; g 2 S; f 6= gg, the pair{set; while (P 6= ;)

hoose (f; g ) 2 P ; P = P r f(f; g)g; h = NF spoly(f; g) j S ; if (h 6= 0) P = P [ f(h; f ) j f 2 S g; S = S [ fhg; return S ;

To see termination of Standard, note that if h 6= 0 then LM(h) 62 L(S ) by property (i) of NF. Hen e, we obtain a stri tly in reasing sequen e of monomial ideals L(S ) of K [x℄, whi h be omes stationary as K [x℄ is Noetherian. That is, after nitely many steps, we always have NF spoly(f; g ) j S = 0 for (f; g ) 2 P , and, again after nitely many steps, the pair-set P will be ome empty. Corre tness follows from applying Bu hberger's fundamental standard basis riterion below.

Remark 1.7.2. If NF is a redu ed normal form and if G is redu ed, then S , as returned by Standard(G,NF), is a redu ed standard basis if we delete elements whose leading monomials are divisible by a leading monomial of another element in S . If G is not redu ed, we may apply a redu ed normal form afterwards to (f; S r ff g) for all f 2 S in order to obtain a redu ed standard basis.

Theorem 1.7.3 (Bu hberger's riterion). Let I R be an ideal and G = fg1 ; : : : ; gs g I. Let NF( j G) be a weak normal form on R with respe t to G. Then the following are equivalent: 7 (1) (2) (3) (4) (5)

G is a standard basis of I. 2 I. Ea h f 2 I has a standard representation with respe t to G. G generates I and NF spoly(gi ; gj ) j G = 0 for i; j = 1; : : : ; s. G generates I and NF spoly(gi ; gj ) j Gij = 0 for a suitable subset Gij G and i; j = 1; : : : ; s.

NF(f j G) = 0 for all f

7 Usually, the impli ation (4) ) (1) is alled Bu hberger's riterion. But with our

on ept of (weak) normal forms, we need, indeed, the impli ation (5) ) (1) to prove the orre tness of the standard basis algorithm.


55

Proof. The impli ation (1) ) (2) follows from Lemma 1.6.7, (2) ) (3) is trivial. To see (3) ) (4), note that h := NF spoly(gi ; gj ) j G 2 I and, hen e, either h = 0 or LM(h) 2 L(G) by (3), a ontradi tion to property (i) of NF. The fa t that G generates I follows immediately from (3). (4) ) (5) is trivial.

Finally, the impli ation (5) ) (1) is the important Bu hberger riterion whi h allows the he king and onstru tion of standard bases in nitely many steps. Our proof uses syzygies and is, therefore, postponed to the next hapter.

Example 1.7.4. Let > be the ordering dp on Mon(x; y), NF=NFBu hberand G = fx2 + y; xy + xg. Then we obtain as initialization S = fx2 + y; xy + xg P = f(x2 + y; xy + x)g.

ger

The while{loop gives, in the rst turn, P =; h = NF( x2 + y2 j S ) = y2 + y P = f(y2 + y; x2 + y); (y2 + y; xy + x)g S = fx2 + y; xy + x; y2 + yg. In the se ond turn P = f(y2 + y; xy + x)g h = NF( x2 y + y3 j S ) = 0. In the third turn P =; h = NF(0 j S ) = 0. The algorithm terminates and S = fx2 + y; xy + x; y 2 + y g is a standard basis.

We present now a general normal form algorithm, whi h works for any monomial ordering. The basi idea is due to Mora [140℄, but our algorithm is more general, with a dierent notion of e art. It has been implemented in Singular sin e 1990, the rst publi ation appeared in [78℄, [90℄. Before going into detail, let us rst analyze Bu hberger's algorithm in the ase of a non{global ordering. We may assume that in K [x; y ℄ we have x1 ; : : : ; xn < 1, y1 ; : : : ; ym > 1 (m 0). Look at the sequen e mi = i xi y i , i 1, of terms onstru ted in the algorithm NFBu hberger. If degx (mi ) is bounded, then, sin e > indu es a well{ordering on K [y ℄, the algorithm stops after nitely many steps. On the other hand, if the degree of mi in x is unbounded, then, for ea h xed fa tor xi , there an only be nitely many ofa tors y j and, hen e, P m onverges in the hxi{adi topology ( f. De nition 6.1.6), that is, Pi1 i i1 mi 2 K [y ℄[[x℄℄. If G = fg1 ; : : : ; gs g we may gather the fa tors mj of any gi , obtaining thus in NFBu hberger an expression

h=f

s X i=1

ai gi ; h; ai 2 K [y℄[[x℄℄ ;

whi h holds in K [y ℄[[x℄℄. However, this pro ess does not stop.

56


The standard example is in one variable x, with x < 1, f := x and G := fg = x x2 g. Using NFBu hberger we obtain 1 ! X xi (x x2 ) = 0 x i=0

i 1=(1 x) in K [[x℄℄. However, the in K [[x℄℄, whi h is true, sin e 1 i=0 xP= i algorithm onstru ts a power series 1 i=0 x having in nitely many terms and not the nite expression 1=(1 x). In order to avoid in nite power series, we have to allow a wider lass of elements for the redu tion in order to reate a standard expression of the form P

uf =

s X i=1

ai gi + NF(f j G) ;

where u is a unit in R, and u, ai and NF(f j G) are polynomials in the ase when the input data f and G = fg1 ; : : : ; gs g are polynomials. In the previous example we arrive at an expression

x)x = x x2

(1 P i instead of x = ( 1 i=0 x )(x

x2 ).

De nition 1.7.5. For f 2 K [x℄ r f0g we de ne the e art of f as e art(f ) := deg f

deg LM(f ) .

Note that, for a homogeneous polynomial f , we have e art(f ) = 0. If w = (w1 ; : : : ; wn ) is any tuple of positive real numbers,we an de ne the weighted e art by e artw (f ) := w{deg(f ) w{deg LM(f ) . In the following normal form algorithm NFMora, we may always take e artw instead of e art, the algorithm works as well. It was noted in [76℄ that, for ertain examples, the algorithm an be ome mu h faster for a good hoi e of w. Another des ription of e art(f ) turns out to be quite useful. Let f h denote the homogenization of f with respe t to a new variable t (su h that all monomials of f are of the same degree, f. Exer ise 1.7.4). De ne on Mon(t; x1 ; : : : ; xn ) an ordering >h by tp x >h tq x if p + jj > q + j j or if p + jj = q + j j and x > x . Equivalently, >h is given by the matrix

01 1 ::: B 0 B ... A 0

11

CC A

where A is a matrix de ning the ordering on K [x℄. This de nes a well{ ordering on Mon(t; x).


For f

57

2 K [x℄ we have LM>h (f h ) = te art(f ) LM> (f ) ;

in parti ular, e art(f ) = degt LM>h (f h ).

Algorithm 1.7.6 (NFMora(f j G)). Let > be any monomial ordering.

Input:

Output:

f

2 K [x℄, G a nite list in K [x℄

h 2 K [x℄ a polynomial weak normal form of f with respe t to G.

h := f ; T := G; while(h 6= 0 and Th := fg 2 T j LM(g ) j LM(h)g 6= ;)

hoose g 2 Th with e art(g ) minimal; if (e art(g ) > e art(h)) T := T [ fhg; h := spoly(h; g); return h;

Example 1.7.7. Let > be the ordering ds on Mon(x; y; z ), f = x2 + y2 + z 3 + x4 + y5, G = fx; yg. Then NFMora (f j G) = z 3 + x4 + y5 . If the input is homogeneous, then the e art is always 0 and NFMora is equal to NFBu hberger. If > is a well{ordering, then LM(g ) j LM(h) implies that LM(g ) LM(h), hen e, even if h is added to T during the algorithm, it annot be used in further redu tions. Thus, NFMora is the same as NFBu hberger, but with a spe ial sele tion strategy for the elements from G.

Proof of Algorithm 1.7.6. Termination is most easily seen by using homogenization: start with h := f h and T := Gh = fg h j g 2 Gg. The while loop looks as follows (see Exer ise 1.7.9):

while (h 6= 0 and Th := fg 2 T j LM(g ) divides t LM(h) for some g 6= ;)

hoose g 2 Th with 0 minimal; if ( > 0) T := T [ fhg; h := spoly(t h; g); h := (hjt=1 )h ; return hjt=1 ;

Sin e R is Noetherian, there exists some positive integer N su h that L(T ) be omes stable for N , where T denotes the set T after the {th turn of the while loop. The next h, therefore, satis es LM(h) 2 L(TN ) = L(T ), when e, LM(g ) divides LM(h) for some g 2 T and = 0. That is, T itself

58


be omes stable for N and the algorithm ontinues with xed T . Then it terminates, sin e > is a well{ordering on K [t; x℄. To see orre tness, onsider the {th turn in the while loop of Algorithm 1.7.6. There we reate (with h0 := f ) h := spoly(h 1 ; g0 ) for some g0 2 T 1 su h that LM(g0 ) j LM(h 1 ). Hen e, there exists some term m 2 K [x℄, LT(m g0 ) = LT(h 1 ), su h that

h = h 1 m g0 ; LM(h 1 ) = LM(m g0 ) > LM(h ) ; Now for g0 we have two possibilities: (1) (2)

g0 = gi 2 G = fg1 ; : : : ; gs g for some i, or g0 2 T r G fh0; h1 ; : : : ; h 2 g.

Suppose, by indu tion, that in the rst stru ted standard representations

uj f =

1 steps (

1) we have on-

s X (j ) i=1

ai gi + hj ; uj 2 S> ; a(ij) 2 K [x℄ ;

0 j 1, starting with u0 := 1, a(0) i := 0. Consider this standard representation for j = 1. In ase (1), we repla e h 1 on the right{hand side by h + m gi , hen e, obtaining

u f =

s X ( ) i=1

ai gi + h

with u := u 1 and some ai( ) 2 K [x℄. In ase (2), we have to substitute h

h + m hj = h with j <

m

1

by

s X (j ) i=1

ai gi

!

uj f

1. Hen e, we obtain an expression (u

1

m uj )f =

s X ( ) i=1

ai gi + h ; ai( ) 2 K [x℄ :

Sin e LM(m ) LM(hj ) = LM(m hj ) = LM(h 1 ) < LM(hj ), we obtain that LM(m ) < 1 and, hen e, u = u 1 m uj 2 S> . It is lear that, with a little extra storage, the algorithm does also return

u 2 S> . Moreover, with quite a bit of bookkeeping one obtains the ai .

Now, the standard basis algorithm for arbitrary monomial orderings looks formally as follows:


59

Algorithm 1.7.8 (StandardBasis(G)).

Let > be any monomial ordering, R = K [x℄> . Input:

Output:

G = fg1; : : : ; gs g K [x℄ S = fh1 ; : : : ; ht g K [x℄ su h that S is a standard basis of the ideal hGiR R.

S = Standard(G,NFMora); return S ;

The following orollary shows that the property of being a standard basis depends only on the ordering of nitely many monomials. This property is used in our study of atness and standard bases (Se tion 7.5).

Corollary 1.7.9 ( nite determina y of standard bases). Let I K [x℄

be an ideal and G K [x℄ be a standard basis of I with respe t to an arbitrary monomial ordering >. Then there exists a nite set F Mon(x) with the following properties: Let >1 be any monomial ordering on Mon(x) oin iding with > on F , then

(1) LM> (g) = LM>1 (g) for all g 2 G, (2) G is a standard basis of I with respe t to >1 . Proof. We apply Theorem 1.7.3 with NF = NFMora. Let G = fg1 ; : : : ; gs g, and let F be the set of all monomials o

urring in all polynomials during the redu tion pro ess of spoly(gi ; gj ) to 0 in NFMora. Then NF spoly(gi ; gj ) j G = 0 also with respe t to >1 , and the result follows, using Theorem 1.7.3 (4).

SINGULAR Example 1.7.10 (standard bases).

The same generators for an ideal give dierent standard bases with respe t to dierent orderings: ring A = 0,(x,y),dp; //global ordering: degrevlex ideal I = x10+x9y2,y8-x2y7; ideal J = std(I); J; //-> J[1℄=x2y7-y8 J[2℄=x9y2+x10 J[3℄=x12y+xy11 //-> J[4℄=x13-xy12 J[5℄=y14+xy12 J[6℄=xy13+y12 ring A1 = 0,(x,y),lp; //global ordering: lex ideal I = fet h(A,I); ideal J = std(I); J; //-> J[1℄=y15-y12 J[2℄=xy12+y14 J[3℄=x2y7-y8 J[4℄=x10+x9y2

60

1. Rings, Ideals and Standard Bases ring B = 0,(x,y),ds; //lo al ordering: lo al degrevlex ideal I = fet h(A,I); ideal J = std(I); J; //-> J[1℄=y8-x2y7 J[2℄=x10+x9y2 ring B1 = 0,(x,y),ls; //lo al ordering: negative lex ideal I = fet h(A,I); ideal J = std(I); J; //-> J[1℄=y8-x2y7 J[2℄=x9y2+x10 J[3℄=x13 intmat O[3℄[3℄=1,1,1,0,-1,-1,0,0,-1; ring C = 0,(t,x,y),M(O); //global ordering: matrix O ideal I = homog(imap(A,I),t); //gives a standard basis for //lo al degrevlex ideal J = std(I); // f. Exer ise 1.7.5 J = subst(J,t,1); J; //-> J[1℄=-x2y7+y8 J[2℄=x9y2+x10 J[3℄=x12y7+x9y10 //already J[1℄,J[2℄ is a //standard basis

We nish this se tion with the so{ alled highest orner, a notion whi h is omputationally extremely useful for 0{dimensional ideals in lo al rings. Moreover, the highest orner is tightly onne ted with the determina y of an isolated hypersurfa e singularity ( f. A.9).

De nition 1.7.11. Let > be a monomial ordering on Mon(x1 ; : : : ; xn ) and

let I K [x1 ; : : : ; xn ℄> be an ideal. A monomial m 2 Mon(x1 ; : : : ; xn ) is

alled the highest orner of I (with respe t to >), denoted by HC(I ), if (1) m 62 L(I ); (2) m0 2 Mon(x1 ; : : : ; xn ), m0 < m =) m0 2 L(I ).

Note that for a global ordering the highest orner is 1 if I is a proper ideal (and does not exist if 1 2 I ). Sin e, by de nition HC(I ) = HC L(I ) , it an be omputed ombinatorially from a standard basis of I . Singular has a built{in fun tion high orner whi h returns, for a given set of generators f1 ; : : : ; fk of I , the highest orner of the ideal hLM(f1 ); : : : ; LM(fk )i, respe tively 0, if the highest orner does not exist.

SINGULAR Example 1.7.12 (highest orner). ring A = 0,(x,y),ds; ideal I = y4+x5,x3y3; high orner(I);


61

//-> // ** I is not a standard basis //-> 0 //no highest orner for std(I); //-> _[1℄=y4+x5 _[2℄=x3y3 high orner(std(I)); //-> x7y2

_[3℄=x8

The highest orner of I is x7 y 2 , as an be seen from Figure 1.2.

y

4

6

L(hy4 + x5 ; x3 y3 i)

F - x

8 4 5 3 3 4 3 3 8 Fig. 1.2. L(hy + x ; x y i) is generated by the monomials y ; x y ; x 7 2 (marked by a ). The highest orner is x y (marked by a F).

Lemma 1.7.13. Let > be a monomial ordering on Mon(x1 ; : : : ; xn ) and I K [x1 ; : : : ; xn ℄> be an ideal. Let m be a monomial su h that m0 < m implies m0 2 L(I ). Let f 2 K [x1 ; : : : ; xn ℄ su h that LM(f ) < m. Then f 2 I. Proof. Let r = NFMora (f j G), G a standard basis for I . If r 6= 0, then LM(r) < LM(f ) < m and, therefore, LM(r) 2 I whi h is a ontradi tion to the properties of the normal form.

Lemma 1.7.14. Let > be a monomial ordering on Mon(x1 ; : : : ; xn ) and denote by z1 ; : : : ; zr the variables < 1 from fx1 ; : : : ; xn g and by y1; : : : ; ys the variables > 1 (0 r; s; r + s = n). Assume that the restri tion of > to Mon(z1 ; : : : ; zr ) is a weighted degree ordering. The following are equivalent for an ideal I K [x1 ; : : : ; xn ℄> : (1) HC(I ) exists, (2) hz1 ; : : : ; zr iN L(I ) for some N 0, (3) hz1 ; : : : ; zr iM I for some M 0.

Moreover, HC(I ) 2 Mon(z1 ; : : : ; zr ) if it exists.

Proof. To see that (1) implies (2), let m := HC(I ). If m = 1, then 1 62 I and z1 ; : : : ; zr 2 L(I ) by de nition of the highest orner. If m 6= 1 and if we write m = xi m0 for some monomial m0 then xi < 1 (otherwise, m0 < m,

62


whi h would imply m0 2 L(I ), hen e, m 2 L(I ), a ontradi tion), and it follows that m 2 Mon(z1 ; : : : ; zr ). Sin e > is a weighted degree ordering on Mon(z1 ; : : : ; zr ), the de nition of the highest orner implies (2). Conversely, if hz1 ; : : : ; zr iN L(I ) then there are only nitely many monomials in Mon(z1 ; : : : ; zr ) whi h are not in L(I ). This nite set has a minimum m. If m0 = z y < m then z < m whi h implies z 2 L(I ) and, hen e, m0 2 L(I ). The impli ation (3) ) (2) being trivial, it remains only to show that (2) implies (3). Let M N . Sin e ziM 2 L(I ), we have ziM + hi 2 I for some hi with LM(hi ) = z y < ziM , in parti ular, z < ziM . Let m := HC(I ), whi h exists by the equivalen e of (2) and (1) proven before, and enlarge M , if ne essary, su h that ziM m. Then z < m implies z 2 L(I ) and, hen e, LM(hi ) 2 L(I ). Now we apply Lemma 1.7.13 and obtain hi 2 I . Therefore, ziM 2 I for i = 1; : : : ; r, and (3) follows.

Remark 1.7.15. As a dire t onsequen e, for a lo al weighted degree ordering,

we have

HC(I ) exists

() dimK (K [x1 ; : : : ; xn ℄> =I ) < 1 () dimK K [x1 ; : : : ; xn ℄=L(I ) < 1 :

Indeed, we show in Se tion 7.5 that, for any monomial ordering,

dimK (K [x1 ; : : : ; xn ℄> =I ) = dimK K [x1 ; : : : ; xn ℄=L(I ) : (see also Corollary 5.3.14).

Remark 1.7.16. The impli ations (2) , (3) ) (1) in Lemma 1.7.14 hold without any assumption on the ordering >. This is a onsequen e of Lemma

1.2.11. The impli ation (1) ) (2) is wrong in general: let > be the negative lexi ographi al ordering ls and I = hxy; x2 i, then HC(I ) = x.

Lemma 1.7.17. Let > be a weighted degree ordering on Mon(x1 ; : : : ; xn ). Moreover, let f1 ; : : : ; fk be a set of generators of the ideal I K [x1 ; : : : ; xn ℄> su h that J := hLM(f1 ); : : : ; LM(fk )i has a highest orner m := HC(J ), and let f 2 K [x1 ; : : : ; xn ℄> . Then the following holds:

(1) HC(I ) exists, and, moreover, HC(I ) HC(J ) and HC(I ) = HC(J ) if f1 ; : : : ; fk is a standard basis of I. (2) If LM(f ) < HC(J ) then f 2 I. (3) For a xed monomial m0 < HC(J ) set M = fi j LM(fi ) m0 g and de ne (

fî :=

fi ; if i 2 M fi + ai m0 ; if i 62 M

where ai 2 K is arbitrary. Then I = hf^1 ; : : : ; f^k i.


63

L(I ) and J = L(I ) if f1 ; : : : ; fk is a standard basis, the

laim follows from Lemma 1.7.14. (2) LM(f ) < m implies LM(f ) 2 L(J ) L(I ). The assertion is a onsequen e of Lemma 1.7.13. (3) Sin e m0 < m, m0 2 I by (2) and, therefore, I = hf^1 ; : : : ; f^k ; m0 i. We have to show m0 2 Ib = hf^1 ; : : : ; f^k i. Sin e LM(fi ) = LM(fî ) for all i, we an apply (2) to Ib instead of I with the same J and m and, therefore, m0 2 Ib.

Proof. (1) Sin e J

The lemma shows that we an delete from fi all terms a m0 , a 2 K , with m0 < minfm; LM(fi )g, still keeping a set of generators of I . This is used in Singular during standard basis omputations in lo al orderings to keep the polynomials sparse and to have early termination if, in the redu tion pro ess, the leading monomial be omes smaller than the highest orner.

Exer ises 1.7.1. Prove the Produ t Criterion : let f; g 2 K [x1 ; : : : ; xn ℄ be polynomials su h that l m LM(f ); LM(g ) = LM(f ) LM(g ), then

NF spoly(f; g ) j ff; g g = 0 : (Hint: It is suÆ ient to prove the statement for NF = NFMora. Assume that LC(f ) = LC(g ) = 1 and laim that spoly(f; g ) = tail(g )f + tail(f )g. Moreover, assume that, after some steps in NFMora, u spoly(f; g ) (u a unit) is redu ed to hf + kg. If LT(hf ) + LT(kg ) = 0 then LT(h) = m LM(g ) and LT(k ) = m LM(f ) for a suitable term m, and (u m) spoly(f; g ) is redu ed to tail(h)f + tail(k )g . If LT(hf ) + LT(kg ) 6= 0 then assume LM(hf + kg ) = LM(hf ), and hf + kg redu es to tail(h)f + kg .)

1.7.2. Let I := hx3 y2 + x4 ; x2 y3 + y4 i K [x; y℄ (resp. I := hx3 + y2 ; y4 + xi).

Compute (without using Singular) a standard basis of I with respe t to the degree lexi ographi al ordering (respe tively lexi ographi al ordering).

1.7.3. Whi h of the following orderings are elimination orderings: (lp(n),ls(m)), (ls(n),lp(m)), (a(1,...,1,0,...0),dp) ?

Compute a standard basis of the ideal hx orderings.

t2 ; y

t3 ; z

lp, ls,

t4 i for all those

1.7.4. For an arbitrary polynomial g 2 K [x1 ; : : : ; xn ℄ of degree d, let gh (x0 ; x1 ; : : : ; xn ) := xd0 g

x x1 ;:::; n x0 x0

2 K [x0 ; : : : ; xn ℄

be the homogenization of g (with respe t to x0 ). For an ideal I let I h := hf h j f 2 I i K [x0 ; : : : ; xn ℄.

K [x1 ; : : : ; xn ℄

64


Let > be a global degree ordering, and let ff1 ; : : : ; fm g be a Grobner basis of I . Prove that

I h = hf1h ; : : : ; fmh i :

1.7.5. For w = (w1 ; : : : ; wn ) 2 Zn, wi 6= 0 for i = 1; : : : ; n, and a polynomial g 2 K [x1 ; : : : ; xn ℄ with w{deg(g) = d, let gh(x0 ; x1 ; : : : ; xn ) := xd0 g

x x1 ; : : : ; wnn w 1 x0 x0

2 K [x0 ; : : : ; xn ℄

be the (weighted) homogenization of g (with respe t to x0 ). For an ideal I K [x1 ; : : : ; xn ℄ let I h := hf h j f 2 I i K [x0 ; : : : ; xn ℄. Let > be a weighted degree ordering with weight ve tor w, and let ff1 ; : : : ; fm g be a Grobner basis of I . Prove that

I h K [x; t℄>h = hf1h ; : : : ; fmh iK [x; t℄>h ; where >h denotes the monomial ordering on Mon(x0 ; : : : ; xn ) de ned by the matrix 0 1 w1 : : : w m 1

B 0 B ...

CC A

A

0

with A 2 GL(n; R) a matrix de ning > on Mon(x1 ; : : : ; xn ). 1.7.6. Let A 2 GL(n; Q ) be a matrix de ning, on Mon(x1 ; : : : ; xn ), the ordering > and let I = hf1 ; : : : ; fm i K [x1 ; : : : ; xn ℄ be an ideal. Consider the ordering >h on Mon(t; x1 ; : : : ; xn ) de ned by the matrix

01 1 ::: B 0 B .. . 0

11

C

C A A

( f. the remark after De nition 1.7.5) and let fG1 ; : : : ; Gs g be a homogeneous h i; f h , the homogenization of f with respe t to standard basis of hf1h ; : : : ; fm i i t. Prove that fG1 jt=1 ; : : : ; Gs jt=1 g is a standard basis for I .

1.7.7. Let I = hf1 ; : : : ; fm i K [x1 ; : : : ; xn ℄ be an ideal, and onsider on

Mon(t; x1 ; : : : ; xn ) the ordering dp (respe tively Dp). Let fG1 ; : : : ; Gs g be a h i; f h , the homogenization of f with respe t to t. standard basis of hf1h ; : : : ; fm i i Prove that fG1 jt=1 ; : : : ; Gs jt=1 g is a standard basis for I with respe t to the ordering ls on Mon(xn ; xn 1 ; : : : ; x1 ) (respe tively Ds on Mon(x1 ; : : : ; xn )).

1.7.8. Let I = hf1 ; : : : ; fm i K [x1 ; : : : ; xn ℄ be an ideal, and onsider on

Mon(x1 ; : : : ; xn ; t) the ordering dp. Let fG1 ; : : : ; Gs g be a standard basis h i, f h the homogenization of f with respe t to t. Prove that of hf1h ; : : : ; fm i i fG1 jt=1 ; : : : ; Gs jt=1 g is a standard basis for I with respe t to the ordering dp on Mon(x1 ; : : : ; xn ).


65

1.7.9. Prove that the while loops in Algorithm 1.7.6 and at the beginning of its proof give the same result.

1.7.10. Che k (by hand) whether the following polynomials f are ontained in the respe tive ideals I :

(1) f = xy 3 z 2 + y 5 z 3 , I = h x3 + y; x2 y z i in Q [x; y; z ℄, (2) f = x3 z 2y 2 , I = hyz y; xy + 2z 2 ; y z i in Q [x; y; z ℄, (3) f and I as in (2) but in Q [x; y; z ℄hx;y;zi .

1.7.11. Verify your omputation in 1.7.10 by using Singular . 1.7.12. Compute a standard basis of (1) hx3 ; x2 y y 3 i with respe t to ls and lp. (2) hx3 + xy; x2 y y 3 i with respe t to ds and dp.

1.7.13. Determine all solutions in C 2 of the system of polynomial equations xy

x 2y + 2 = 0 ; x2 + xy

2x = 0 :

(Hint: ompute rst a lexi ographi al Grobner basis of the two polynomials.)

1.7.14. Use Singular to determine all points in C 3 lying on the variety V given by:

(1) V = V (xz y; xy + 2z 2 ; y z ), (2) V = V (x2 + y 2 + z 2 1; y 2 z; x2 + y 2 ).

1.7.15. Consider f (x; y) := x2 y3

3 2 2y .

(1) Compute all riti al points of f (that is, points where f=x and f=y vanish). (2) Whi h of the riti al points are lo al minima, maxima, saddle points? (3) Do the same for g (x; y ) = f (x; y ) (y 1).

1.7.16. Let K be a eld, let m K [x1 ; : : : ; xn ℄ be a maximal ideal, and let

L := K [x1 ; : : : ; xn ℄=m. Moreover, let I = hf1 ; : : : ; fmi L[y1 ; : : : ; ys ℄ be an ideal, and let J := hF1 ; : : : ; Fm i K [x1 ; : : : ; xn ; y1 ; : : : ; ys ℄ be generated by representatives Fi of the fi , i = 1; : : : ; m. Finally, let fH1 ; : : : ; Ht g be a standard basis of J with respe t to a blo k ordering >= (>1 ; >2 ) on Mon(y1 ; : : : ; ys ; x1 ; : : : ; xn ) with >1 ; >2 global. (1) Prove that fH1 mod m; : : : ; Ht mod mg is a standard basis of I with respe t to >1 . (2) Write a Singular pro edure to ompute a minimal standard basis in L[y1; : : : ; ys ℄ (where K is one of the base elds of Singular ) su h that the leading oeÆ ients are 1.

66


1.7.17. Let I K [x1 ; : : : ; xn ℄ be an ideal and > a monomial ordering. Then

there exists a weight ve tor w = (w1 ; : : : ; wn ) 2 Zn, with wi > 0 if xi > 1 and wi < 0 if xi < 1, su h that the weighted degree lexi ographi al ordering de ned by w and the given ordering > yield the same leading ideal L(I ). (Hint: use Lemma 1.2.11.)

1.7.18. (1) Let I K [x1 ; : : : ; xn ℄> be an ideal, and let > denote the negative lexi ographi al ordering ls. Moreover, let x , = (1 ; : : : ; n ) denote the highest orner of I . Show that, for i = 1; : : : ; n,

i = max p x1 1 : : : xii 11 xpi 62 L(I ) : (2) Compute the highest orner of I = hx2 + x2 y; y 3 + xy 3 ; z 3 xz 2 i with respe t to the orderings ls and ds. (This an be done by hand; you may he k your results by using the Singular fun tion high orner.)

1.7.19. Let K be a eld, x one variable and > the well{ordering on K [x℄. (1) Prove that the standard basis algorithm is the Eu lidean algorithm. (2) Use Singular to ompute for f = (x3 + 5)2 (x 2)(x2 + x + 2)4 and g = (x3 + 5)(x2 3)(x2 + x + 2) the g d(f; g). Try std(ideal(f,g)) and g d(f,g).

1.7.20. Let K be a eld, x = (x1 ; : : : ; xn ) and > the lexi ographi al ordering

on K [x℄.

(1) Prove that the standard basis algorithm is the Gaussian elimination algorithm if it is applied to linear polynomials. (2) Use Singular to solve the following linear system of equations: 22x + 77y + z = 3 x + y + z = 77 x y z = 11 : With option(redSB) the omplete redu tion of the standard basis an be for ed. Try both possibilities.

1.7.21. Prove that the equivalen e of (2) and (3) in Lemma 1.7.14 holds for any monomial ordering. 1.7.22. Let > be an arbitrary monomial ordering on Mon(x1 ; : : : ; xn ), and let I K [x℄ be an ideal. Let G K [x℄ be a standard basis of I with respe t to >. Assume, moreover that dimK (K [x℄=L(I )) < 1. Prove that there exists a standard basis G0 G su h that RedNFBu hberger( j G0 ) terminates. (Hint: Denote by z = (z1 ; : : : ; zr ) the variables < 1. Use Exer ise 1.7.21 to

hoose M su h that hz iM I . Enlarge G by adding all monomials in z of degree M .)

1.8 Operations on Ideals and their Computation

67

1.7.23. Let > be an arbitrary monomial ordering on Mon(x1 ; : : : ; xn ), and let I K [x℄> be an ideal. Denote by z = (z1 ; : : : ; zr ) the variables < 1, and assume that hz im I for some positive integer m. Prove that the anoni al inje tion K [x℄=(I \ K [x℄) ,! K [x℄> =I is an isomorphism. 1.7.24. Use Remark 1.6.14 and Exer ise 1.6.9 for writing a Singular pro edure whi h omputes standard bases over a polynomial ring K [t1 ; : : : ; ts ℄, K a eld.

1.8 Operations on Ideals and their Computation The methods developed so far already allow some interesting appli ations to basi ideal operations. In general, we assume we have given a nite set of ideals, ea h is given by a nite set of polynomial generators. We want to either aÆrmatively answer a spe i question about the ideals or to ompute a spe i operation on these ideals, that is, ompute a nite set of generators for the result of the operation.

1.8.1 Ideal Membership Let K [x℄ = K [x1 ; : : : ; xn ℄ be the polynomial ring over a eld K , >0 an arbitrary monomial ordering and R = K [x℄>0 the ring asso iated to K [x℄ and >0 . Re all that K [x℄ R K [x℄hxi , and that R = K [x℄hxi if and only if >0 is lo al ( f. Se tion 1.5). Let NF denote a weak normal form and redNF a redu ed normal form ( f. Se tion 1.6). We do not need any further assumptions about NF, respe tively redNF, however, we may think of NFBu hberger (1.6.10), respe tively redNFBu hberger (1.6.11), if >0 is global, and NFMora (1.7.6) in the general ase. These are also the normal forms implemented in Singular.

Problem: Given f; f1; : : : ; fk 2 K [x℄, and let I = hf1 ; : : : ; fk iR . We wish to de ide whether f 2 I , or not. Solution: We hoose any monomial ordering > su h that K [x℄> = R and

ompute a standard basis G = fg1 ; : : : ; gs g of I with respe t to >. If NF is any weak normal form, then f 2 I if and only if NF(f j G) = 0. Corre tness follows from Lemma 1.6.7.

Sin e the result is independent of the hosen NF, we should use, for reasons of eÆ ien y, a non{redu ed normal form. If >0 is global, we usually hoose dp and, if >0 is lo al, then ls or ds are preferred.

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SINGULAR Example 1.8.1 (ideal membership). (1) Che k in lusion of a polynomial in an ideal

ring A = 0,(x,y),dp; ideal I = x10+x9y2,y8-x2y7; ideal J = std(I); poly f = x2y7+y14; redu e(f,J,1); //3rd parameter 1 avoids tail redu tion //-> -xy12+x2y7 //f is not in I f = xy13+y12; redu e(f,J,1); //-> 0 //f is in I

(2) Che k in lusion and equality of ideals. ideal K = f,x2y7+y14; redu e(K,J,1); //normal form for ea h generator of K //-> _[1℄=0 _[2℄=-xy12+x2y7 K=f,y14+xy12; size(redu e(K,J,1));

//K is not in I //result is 0 iff K is in I

//-> 0

Now assume that f 2 I = hf1 ; : : : ; fk iR . Then there exist u 2 K [x℄ \ R , a1 ; : : : ; ak 2 K [x℄ su h that

uf = a1 f1 + + ak fk :

(*)

If ff1 ; : : : ; fk g is a standard basis of I , then, in prin iple, the normal form algorithm NFMora provides u and the ai . However, it is also possible to express f as a linear ombination of arbitrary given generators f1 ; : : : ; fk , by using the lift or division ommand. How this an be done is explained in Chapter 2, Se tion 2.8.1. If the ordering is global, then we an hoose u = 1 in the above expression (*). This is illustrated in the following example.

SINGULAR Example 1.8.2 (linear ombination of ideal members). We exemplify the Singular ommands lift and division:

ring A = 0,(x,y),dp; ideal I = x10+x9y2,y8-x2y7; poly f = xy13+y12; matrix M=lift(I,f); //f=M[1,1℄*I[1℄+...+M[r,1℄*I[r℄ M; //-> M[1,1℄=y7 //-> M[2,1℄=x7y2+x8+x5y3+x6y+x3y4+x4y2+xy5+x2y3+y4


69

Hen e, f an be expressed as a linear ombination of I [1℄ and I [2℄ using M : f-M[1,1℄*I[1℄-M[2,1℄*I[2℄; //test //-> 0

In a lo al ring we an, in general, only express uf as a polynomial linear

ombination of the generators of I if f 2 I : ring R = 0,(x,y,z),ds; poly f = yx2+yx; ideal I = x-x2,y+x; list L = division(f,I); //division with remainder L; //-> [1℄: [2℄: [3℄: //-> _[1,1℄=y-y2 _[1℄=0 _[1,1℄=1+y //-> _[2,1℄=2xy matrix(I)*L[1℄ - matrix(f)*L[2℄ - matrix(L[3℄); //-> _[1,1℄=0

Hen e (1 + y )f = (x

x2 )(y

//test

y2 ) + (y + x)(2xy), the remainder being 0.

1.8.2 Interse tion with Subrings (Elimination of variables) This is one of the most important appli ations of Grobner bases. The problem may be formulated as follows (we restri t ourselves for the moment to the

ase of the polynomial ring):

2 K [x℄ = K [x1 ; : : : ; xn ℄, I = hf1 ; : : : ; fk iK [x℄, we should like to nd generators of the ideal

Problem: Given f1 ; : : : ; fk

I 0 = I \ K [xs+1 ; : : : ; xn ℄; s < n : Elements of the ideal I 0 are said to be obtained from f1 ; : : : ; fk by eliminating x1 ; : : : ; x s .

In order to treat this problem, we need a global elimination ordering for

x1 ; : : : ; xs . We an use the lexi ographi al ordering lp whi h is an elimination ordering (De nition 1.5.4) for ea h s, but lp is, in almost all ases, the most

expensive hoi e. A good hoi e is, usually, (dp(s),dp(n-s)), the produ t ordering of two degrevlex orderings. But there is another way to onstru t an elimination ordering whi h is often quite fast. Let > be an arbitrary ordering and let a1 ; : : : ; as be positive integers. De ne >a by

x >a x :() a1 1 + + as s > a1 1 + + as s or a1 1 + + as s = a1 1 + + as s and x > x :

70


Then >a is an elimination ordering and a = (a1 ; : : : ; as ) is alled an extra weight ve tor . If > is an arbitrary elimination ordering for x1 ; : : : ; xs , then

K [x1 ; : : : ; xn ℄> = (K [xs+1 ; : : : ; xn ℄>0 )[x1 ; : : : ; xs ℄ ; sin e the units in K [x℄> do not involve x1 ; : : : ; xs (we denote, by >0 , the ordering on Mon(xs+1 ; : : : ; xn ) indu ed by >). Hen e, f 2 K [xs+1 ; : : : ; xn ℄>0 for any f 2 K [x1 ; : : : ; xn ℄> su h that LM(f ) 2 K [xs+1 ; : : : ; xn ℄. The following lemma is the basis for solving the elimination problem.

Lemma 1.8.3. Let > be an elimination ordering for x1 ; : : : ; xs on the set of monomials Mon(x1 ; : : : ; xn ), and let I K [x1 ; : : : ; xn ℄> be an ideal. If S = fg1; : : : ; gk g is a standard basis of I, then S 0 := fg 2 S j LM(g) 2 K [xs+1 ; : : : ; xn ℄g

is a standard basis of I 0 := I \ K [xs+1 ; : : : ; xn ℄>0 . In parti ular, S 0 generates the ideal I 0 . Proof. Given f 2 I 0 I there exists gi 2 S su h that LM(gi ) divides LM(f ), sin e S is a standard basis of I . Sin e f 2 K [xs+1 ; : : : ; xn ℄> , we have LM(f ) 2 K [xs+1 ; : : : ; xn ℄ and, hen e, gi 2 S 0 by the above remark. Finally, sin e S 0 I 0 , S 0 is a standard basis of I 0 . The general elimination problem an be posed, for any ring asso iated to a monomial ordering, as follows. Re all that the ordering on the variable to be eliminated must be global.

Problem: Given polynomials f1; : : : ; fk 2 K [x1 ; : : : ; xn ℄, let I := hf1 ; : : : ; fk iR with R := (K [xs+1 ; : : : ; xn ℄> )[x1 ; : : : ; xs ℄ for some monomial ordering > on Mon(xs+1 ; : : : ; xn ). Find generators for the ideal I 0 := I \ K [xs+1 ; : : : ; xn ℄> .

Solution: Choose an elimination ordering for x1 ; : : : ; xs on Mon(x1 ; : : : ; xn ), whi h indu es the given ordering > on Mon(xs+1 ; : : : ; xn ), and ompute a standard basis S = fg1 ; : : : ; gk g of I . By Lemma 1.8.3, those gi , for whi h LM(gi ) does not involve x1 ; : : : ; xs , generate I 0 (even more, they are a standard basis of I 0 ). A good hoi e of an ordering on Mon(x1 ; : : : ; xn ) may be (dp(s),>), but instead of > we may hoose any ordering >0 on Mon(xs+1 ; : : : ; xn ) su h that K [xs+1 ; : : : ; xn ℄>0 = K [xs+1 ; : : : ; xn ℄> . For any global ordering > on Mon(xs+1 ; : : : ; xn ), we have, thus, a solution to the elimination problem in the polynomial ring, as stated at the beginning of this se tion.


71

SINGULAR Example 1.8.4 (elimination of variables). ring A =0,(t,x,y,z),dp; ideal I=t2+x2+y2+z2,t2+2x2-xy-z2,t+y3-z3; eliminate(I,t); //-> _[1℄=x2-xy-y2-2z2

_[2℄=y6-2y3z3+z6+2x2-xy-z2

Alternatively hoose a produ t ordering: ring A1=0,(t,x,y,z),(dp(1),dp(3)); ideal I=imap(A,I); ideal J=std(I); J; //-> J[1℄=x2-xy-y2-2z2 J[2℄=y6-2y3z3+z6+2x2-xy-z2 //-> J[3℄=t+y3-z3

We an also hoose the extra weight ve tor a = (1; 0; 0; 0) to obtain an elimination ordering: ring A2=0,(t,x,y,z),(a(1),dp); ideal I=imap(A,I); ideal J=std(I); J; //-> J[1℄=x2-xy-y2-2z2 J[2℄=y6-2y3z3+z6+2x2-xy-z2 //-> J[3℄=t+y3-z3

By Lemma 1.8.3, the elements of J whi h do not involve t (here J[1℄ and J[2℄), are a standard basis of I \ K [x; y; z ℄.

1.8.3 Zariski Closure of the Image Here we study the geometri ounterpart of elimination. The reader who is not familiar with the geometri al ba kground should read Se tion A.1 rst. In this se tion we assume K to be algebrai ally losed. Suppose ' : K [x℄ = K [x1 ; : : : ; xn ℄ ! K [t℄ = K [t1 ; : : : ; tm ℄ is a ring map given by f1 ; : : : ; fn 2 K [t℄ su h that '(xi ) = fi . Let I = hg1 ; : : : ; gk i K [t℄ and J = hh1 ; : : : ; hl i K [x℄ be ideals su h that '(J ) I . Then ' indu es a ring map ' : K [x℄=J ! K [t℄=I and, hen e, we obtain a ommutative diagram of morphism of aÆne s hemes ( f. Se tion A.1)

X := V (I )

f ='#

V (J ) =: Y /

_

A

m

_

'#

/

A

n

:

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We annot ompute the image f (X ), sin e it is, in general, not losed. However, we an ompute the (Zariski) losure f (X ).

Problem: The problem is to nd polynomials p1 ; : : : ; pr 2 K [x℄ su h that f (X ) = V (p1 ; : : : ; pr ) A n : Solution: De ne the ideal N = hh1 ; : : : ; hl ; g1(t); : : : ; gk (t); x1 f1(t); : : : ; xn fn (t)iK [t;x℄

and eliminate t1 ; : : : ; tm from N , that is, ompute generators p1 ; : : : ; pr 2 K [x℄ of N \ K [x℄. Then V (p1 ; : : : ; pr ) = f (X ). Hen e, we an pro eed as in Se tion 1.8.2. We hoose a global ordering whi h is an elimination ordering for t1 ; : : : ; tm on Mon(t1 ; : : : ; tm ; x1 ; : : : ; xn ),

ompute a Grobner basis G of N and sele t those elements p1 ; : : : ; pr from G whi h do not depend on t. Corre tness follows from Lemma A.3.10.

Sin e K is algebrai ally losed (with f = (f1 ; : : : ; fn ) : K m ! K n ), Lemma A.2.18 implies

f (X ) = fx 2 K n j 9 t 2 K m su h that f (t) = xg : The following example shows that the question whether f (X ) is losed or not may depend on the eld.

Example 1.8.5. Consider the ring map ' : K [x℄ ! K [x; y℄=hx2 + y2 1i given by '(x) := x, and the indu ed morphism X := V (hx2 + y2 _

1i)

A2

f ='# '#

/

V (h0i) /

A1 :

It is easy to see that, if K is algebrai ally losed, then f is surje tive, and hen e, f (X ) is losed. However, if K = R , then f (X ) is a segment but the Zariski losure of the segment is the whole line. Now we treat the problem of omputing the losure of the image of a map between spe tra of lo al rings. More generally, let >1 , respe tively >2 , be monomial orderings on Mon(x1 ; : : : ; xn ), respe tively Mon(t1 ; : : : ; tm ), and let ' : K [x℄>1 ! K [t℄>2 be a ring map de ned by '(xi ) = fi (t) 2 K [t℄ ( f. Lemma 1.5.8). Let I K [t℄ and J K [x℄ be ideals as above, satisfying '(J ) I , and

' : K [x℄>1 =J the indu ed map.

! K [t℄>2 =I


73

Problem: We want to ompute equations for f (X ) Y for the map f = '# : X = Spe (K [t℄>2 =I )

!Y

= Spe (K [x℄>1 =J ) :

We laim that the following algorithm solves the problem:

Solution: Choose any ordering > on Mon(t1 ; : : : ; tm ; x1 ; : : : ; xn ) whi h is an elimination ordering for t1 ; : : : ; tm and satis es K [x℄>0 K [x℄>1 where >0 is the ordering on Mon(x1 ; : : : ; xn ) indu ed by >.8 Compute a standard basis G of the ideal N := hI; J; x1 f1(t); : : : ; xn fn(t)i as above with respe t to this ordering. Sele t those elements p1 ; : : : ; pr from G whi h do not depend on t. Then f (X ) = V (hp1 ; : : : ; pr iK [x℄>1 ).

The only problem in seeing orre tness results from the fa t that we only assume K [x℄>0 K [x℄>1 but no other relation between > and >1 ; >2 . The graph onstru tion from Appendix A.2, applied to the lo alized rings, shows that f (X ) is the zero{set of the ideal

N (K [t℄>2 K K [x℄>1 ) \ K [x℄>1 : Now the above algorithm omputes polynomial generators of the interse tion (N K [t; x℄> ) \ K [x℄>0 . We have K [t; x℄> = K [t℄ K [x℄>0 , K [x℄>0 K [x℄>1 and an in lusion of rings

K [t; x℄ R1 := K [t; x℄> R2 := K [t℄>2 K [x℄>1

R3 := K [t; x℄(>2 ;>1 ) ;

where (>2 ; >1 ) is the produ t ordering on Mon(t1 ; : : : ; tm ; x1 ; : : : ; xn ). Moreover, by Lemma 1.4.8 (1), we have (N R3 ) \ K [t; x℄ = N , hen e, (N Ri ) \ K [t; x℄ = N for i = 1; 2 and, therefore, (N R1 ) \ K [x℄ = N \ K [x℄ = (N R2 ) \ K [x℄ : Again, by Lemma 1.4.8 (1), (N R2 ) \ K [x℄>1 = (N R2 \ K [x℄) K [x℄>1 and (N R1 ) \ K [x℄>0 = (N R1 \ K [x℄) K [x℄>0 . Altogether, we have

(N R2 ) \ K [x℄>1 = (N R1 \ K [x℄) K [x℄>1 = (N R1 ) \ K [x℄>0 K [x℄>1 ; where the left{hand side de nes f (X ) and generators for the right{hand side are omputed. Thus, we have many hoi es for orderings on Mon(x1 ; : : : ; xn ) for omputing f (X ) Y . In parti ular, we an always hoose a global ordering. 8 We ould hoose, for example, > to be dp(m); dp(n) or > to be dp(m); >1 .

74


SINGULAR Example 1.8.6 (Zariski losure of the image).

Compute an impli it equation for the surfa e de ned parametri ally by the map f : A 2 ! A 3 , (u; v ) 7! (uv; uv 2 ; u2 ). ring A =0,(u,v,x,y,z),dp; ideal I=x-uv,y-uv2,z-u2; ideal J=eliminate(I,uv); J; //-> J[1℄=x4-y2z //defines the losure of f(X)

Note that the image does not ontain the y {axis, however, the losure of the image ontains the y {axis. This surfa e is alled the Whitney umbrella .

Fig. 1.3.

Whitney Umbrella

1.8.4 Solvability of Polynomial Equations Problem: Given f1 ; : : : ; fk 2 K [x1 ; : : : ; xn ℄, we want to assure whether the system of polynomial equations f1 (x) = = fk (x) = 0 n

has a solution in K , where K is the algebrai losure of K . Let I = hf1 ; : : : ; fk iK [x℄ , then the question is whether the algebrai set n V (I ) K is empty or not. Solution: By Hilbert's Nullstellensatz, V (I ) = ; if and only if 1 2 I . We

ompute a Grobner basis G of I with respe t to any global ordering on Mon(x1 ; : : : ; xn ) and normalize it (that is, divide every g 2 G by LC(g )). Sin e 1 2 I if and only if 1 2 L(I ), we have V (I ) = ; if and only if 1 is an element of a normalized Grobner basis of I . Of ourse, we an avoid normalizing, whi h is expensive in rings with parameters. Sin e 1 2 I if and only if G ontains a non{zero onstant polynomial, we have only to look for an element of degree 0 in G.


75

1.8.5 Solving Polynomial Equations A fundamental task with ountless appli ations is to solve a system of polynomial equations, f1 (x) = 0; : : : ; fk (x) = 0, fi 2 K [x℄ = K [x1 ; : : : ; xn ℄. However, what is a tually meant by \solving" very mu h depends on the ontext. For instan e, it ould mean to determine one (respe tively some, respe tively all) points of the solution set V (f1 ; : : : ; fk ), either onsidered as a subset of n K n or of K , where K is the algebrai losure of K (for notations f. Se tion A.1). Here, we onsider only the ase where the ideal I = hf1 ; : : : ; fk iK [x℄ is 0{ dimensional, that is, where f1 = = fk = 0 has only nitely many solutions n in K . T From an algebrai point of view, a primary de omposition I = ri=1 Qi p of I with Pi = Qi K [x1 ; : : : ; xn ℄ a maximal ideal, ould be onsidered as a solution ( f. Chapter 4). At least, it provides a de omposition V (I ) = V (P1 ) [ [ V (Pr ) and if p = (p1 ; : : : ; pn ) 2 K n is a solution, then Pj = hx1 p1 ; : : : ; xn pn i for some j and we an, indeed, re over the oordinates n of p from a primary de omposition of I . Moreover, for solutions p 2 K whi h n are not in K the primary de omposition provides irredu ible polynomials e of K su h that p has oordinates in K e ( f. Chapter de ning a eld extension K 4). Besides the fa t that primary de omposition is very expensive, the answer would be unsatisfa tory from a pra ti al point of view. Indeed, if K = R or C , most people would probably interpret solving as nding approximate numeri al oordinates of one (respe tively some, respe tively all) point(s) of V (I ). And this means that, at some point, we need a numeri al root nder. Numeri al solving of equations (even trans endental or (partial) dierential equations) is a highly developed dis ipline in mathemati s whi h is very su

essful in appli ations to real life problems. However, there are inherent problems whi h often make it diÆ ult, or even impossible, either to nd a solution or to ensure that a dete ted solution is (approximately) orre t. Parti ular problems are, for example, to nd all solutions or to guarantee stability and onvergen e of algorithms in the presen e of singularities. In this ontext symboli methods an be useful in preparing the system by nding another set of generators for I (hen e, having the same solutions) whi h is better suited for numeri al omputations. Here we des ribe only how lexi ographi al Grobner bases an be used to redu e the problem of multivariate solving to univariate solving.

Problem: Given f1 ; : : : ; fk 2 K [x1 ; : : : ; xn ℄, K = R or C , whi h we assume to have only nitely many solutions p1 ; : : : ; pr 2 C n . We wish to nd oordinates of all pi in de imal format up to a given number of digits. We are also interested in lo ating multiple solutions. Solution: Compute a lexi ographi al Grobner basis G = fg1 ; : : : ; gs g of I for x1 > x2 > > xn . Then we have ( f. Exer ise 1.8.6) s n and, after

76


renumbering G, there are elements g1 ; : : : ; gn 2 G su h that

g1 = g1 (xn ); LM(g1 ) = xnnn ; g2 = g2 (xn 1 ; xn ); LM(g2 ) = xnnn 11 ; .. .

gn = gn (x1 ; : : : ; xn ); LM(gn ) = xn1 1 : Now use any numeri al univariate solver (for example Laguerre's method) to nd all omplex solutions of g1 (xn ) = 0 up to the required number of digits. Substitute these in g2 and for ea h substitution solve g2 in xn 1 , as before. Continue in this way up to gn . Thus, we omputed all oordinates of all solutions of g1 = = gn = 0. Finally, we have to dis ard those solutions for whi h one of the remaining polynomials gn+1 ; : : : ; gs does not vanish. We should like to mention that this is not the best possible method. In parti ular, the last step, that is, dis arding non{solutions, may lead to numeri al problems. A better method is to use triangular sets , either in the spirit of Lazard [120℄ or Moller [135℄ ( f. [82℄ for experimental results and a omparison to resultant based methods). Triangular sets are implemented in the Singular library triang.lib.

SINGULAR Example 1.8.7 (solving equations). ring A=0,(x,y,z),lp; ideal I=x2+y+z-1, x+y2+z-1, x+y+z2-1; ideal J=groebner(I); //the lexi ographi al Groebner basis J; //-> J[1℄=z6-4z4+4z3-z2 J[2℄=2yz2+z4-z2 //-> J[3℄=y2-y-z2+z J[4℄=x+y+z2-1

We use the multivariate solver based on triangular sets, due to Moller and Hillebrand [135℄, [100℄, and the univariate Laguerre{solver. LIB"solve.lib"; list s1=solve(I,6); //-> // name of new urrent ring: AC s1; //-> [1℄: [2℄: [3℄: [4℄: [5℄: //-> [1℄: [1℄: [1℄: [1℄: [1℄: //-> 0.414214 0 -2.414214 1 0 //-> [2℄: [2℄: [2℄: [2℄: [2℄: //-> 0.414214 0 -2.414214 0 1 //-> [3℄: [3℄: [3℄: [3℄: [3℄: //-> 0.414214 1 -2.414214 0 0


77

If we want to ompute the zeros with multipli ities then we use 1 as a third parameter for the solve ommand: setring A; list s2=solve(I,6,1); s2; //-> [1℄: [2℄: //-> [1℄: [1℄: //-> [1℄: [1℄: //-> [1℄: [1℄: //-> -2.414214 0 //-> [2℄: [2℄: //-> -2.414214 1 //-> [3℄: [3℄: //-> -2.414214 0 //-> [2℄: [2℄: //-> [1℄: [1℄: //-> 0.414214 1 //-> [2℄: [2℄: //-> 0.414214 0 //-> [3℄: [3℄: //-> 0.414214 0 //-> [2℄: [3℄: //-> 1 [1℄: //-> 0 //-> [2℄: //-> 0 //-> [3℄: //-> 1 //-> [2℄: //-> 2

The output has to be interpreted as follows: there are two zeros of multipli ity 1 and three zeros ((0; 1; 0), (1; 0; 0), (0; 0; 1)) of multipli ity 2. Note that a possible way to he k whether a system of polynomial equations f1 = = fk = 0 has nitely many solutions in K , is to ompute a Grobner basis G of I = hf1 ; : : : ; fk i with respe t to any ordering (usually dp is the fastest). Then V (I ) is nite if and only if dim(G)=0 or, equivalently, lead(G) ontains xni i for i = 1; : : : ; n and some ni (and then the number of solutions is ni : : : nn ). The number of solutions, ounting multipli ities, n in K is equal to vdim(G)= dimK K [x℄=I ( f. Exer ises 1.8.6 to 1.8.8).

1.8.6 Radi al Membership Problem : Let f1 ; : : : ; fk 2 K [x℄> , > a monomial ordering on Mon(x1 ; : : : ; xn ) and I = hf1 ; : : : ; fk iK [x℄> . Given some f 2 K [x℄> we want to de ide whether

78

p


2 I . The following lemma, whi h is sometimes alled Rabinowi h's tri k, is the basis for solving this problem. 9

f

Lemma 1.8.8. Let A be a ring, I A an ideal and f 2 A. Then f

2

p

I

() 1 2 I~ := hI; 1

tf iA[t℄

where t is an additional new variable. Proof. If f m 2 I then tm f m 2 I~ and, hen e, 1 = tm f m + (1

tm f m ) = tm f m + (1 tf )(1 + tf + + tm 1 f m 1) 2 I~ :

Conversely, let 1 2 I~. Without loss of generality, p we may assume that f is not nilpotent sin e, otherwise, f is learly in I . P i By assumption, there are f1 ; : : : ; fk 2 I and ai (t) = dj =0 aij tj 2 A[t℄, i = 0; : : : ; k su h that 1=

k X i=1

ai (t)fi + a0 (t)(1 tf ) :

Sin e f is not nilpotent we an repla e t by 1=f and obtain 1=

X

i

ai

1

f

fi =

X

i;j

aij f j fi

in the lo alization Af , see Multiplying with f m , for m suÆ iently P Se tionm1.4. m j large, we obtain f = i;j (aij f )fi 2 I (even in A, not only in Af ).

Solution : By Lemma 1.8.8, we have f

2

1 2 J := hf1 ; : : : ; fk ; 1

p

I if and only if tf i(K [x℄> )[t℄ ;

where t is a new variable. To solve the problem, we hoose on Mon(t; x1 ; : : : ; xn ) an elimination ordering for t indu ing >0 on Mon(x1 ; : : : ; xn ) su h that K [x℄>0 = K [x℄> (for p example, take (lp(1),>)) and ompute a standard basis G of J . Then f 2 I if and only if G ontains an element g with LM(g) = 1.

SINGULAR Example 1.8.9 (radi al membership). ring A =0,(x,y,z),dp; ideal I=x5,xy3,y7,z3+xyz; poly f =x+y+z;

9 We an even ompute the full radi al pI as is shown in Se tion 4.5, but this is a mu h harder omputation.


79

ring B =0,(t,x,y,z),dp; //need t for radi al test ideal I=imap(A,I); poly f =imap(A,f); I=I,1-t*f; std(I); //-> _[1℄=1 //f is in the radi al LIB"primde .lib"; //just to see, we ompute the radi al setring A; radi al(I); //-> _[1℄=z _[2℄=y _[3℄=x

1.8.7 Interse tion of Ideals Problem: Given f1 ; : : : ; fk , h1 ; : : : ; hr 2 K [x℄ and > a monomial ordering. Let I1 = hf1 ; : : : ; fk iK [x℄> and I2 = hh1 ; : : : ; hr iK [x℄> . We wish to nd generators for I1 \ I2 . Consider the ideal J := htf1 ; : : : ; tfk ; (1 t)h1 ; : : : ; (1 t)hr i(K [x℄> )[t℄.

Lemma 1.8.10. With the above notations, I1 \ I2 = J \ K [x℄> . Proof. Let f

2 J \ K [x℄> , then

f (x) = t

k X i=1

ai (t; x)fi (x) + (1 t)

r X j =1

bj (t; x)hj (x) : P

Sin e theP polynomial f is independent of t, we have f = ki=1 ai (1; x)fi 2 I1 and f = rj=1 bj (0; x)hj 2 I2 , hen e f 2 I1 \ I2 . Conversely, if f 2 I1 \ I2 , then f = tf + (1 t)f 2 J \ K [x℄> .

Solution: We hoose an elimination ordering for t on Mon(t; x1 ; : : : ; xn ) indu ing >0 on Mon(x1 ; : : : ; xn ) su h that K [x℄>0 = K [x℄> (for example, take (lp(1),>)). Then we ompute a standard basis of J and get generators for J \ K [x℄> as in Se tion 1.8.2 A dierent solution, using syzygies, is des ribed in Chapter 2, Se tion 2.8.3.

SINGULAR Example 1.8.11 (interse tion of ideals). ring A=0,(x,y,z),dp; ideal I1=x,y; ideal I2=y2,z; interse t(I1,I2); //the built-in SINGULAR ommand //-> _[1℄=y2 _[2℄=yz _[3℄=xz ring B=0,(t,x,y,z),dp;

//the way des ribed above

80

1. Rings, Ideals and Standard Bases ideal I1=imap(A,I1); ideal I2=imap(A,I2); ideal J=t*I1+(1-t)*I2; eliminate(J,t); //-> _[1℄=yz _[2℄=xz _[3℄=y2

1.8.8 Quotient of Ideals Problem: Let I1 and I2 K [x℄> be as in Se tion 1.8.7. We want to ompute I1 : I2 = fg 2 K [x℄> j gI2 I1 g : T

Sin e, obviously, I1 : hh1 ; : : : ; hr i = ri=1 (I1 : hhi i), we an ompute I1 : hhi i for ea h i and then apply Singular Example 1.8.11. The next lemma shows a way to ompute I1 : hhi i.

Lemma 1.8.12. Let I K [x℄> be an ideal, and let h 2 K [x℄> , h 6= 0. Moreover, let I \ hhi = hg1 h; : : : ; gs hi. Then I : hhi = hg1 ; : : : ; gs iK [x℄> .

Proof. Any set of generators of I \ hhi is of the form fg1h; : : : ; gs hg. Therefore, hhg1 ; : : : ; gs i I , hen e hgP 1 ; : : : ; gs i I : hhi. Conversely, if g 2 I : hhi, then hg 2 I \ hhi and hg = h P i ai gi for some ai . Sin e K [x℄> has no zerodivisors and h 6= 0, we have g = i ai gi whi h proves the laim.

Solution 1 : We an ompute I1 : I2 by omputing, for i = 1; : : : ; r, I1 \ hhi i a

ording to Se tion 1.8.7,T divide the generators by hi getting I1 : hhi i and

ompute the interse tion i (I1 : hhi i), a

ording to Se tion 1.8.7. T

hhi i), we an de ne h := h1 + t1 h2 + + tr 1 hr 2 K [t1 ; : : : ; tr

Instead of omputing

i (I1 :

1 ; x1 ; : : : ; x n ℄

and obtain

I1 : I2 = (I1 : hhi) \ K [x℄> :

This holds, sin e g (x) 2 I1 : hhi if and only if

g(x) h1 (x) + t1 h2 (x) + + tr 1 hr (x) =

k X i=1

ai (x; t)fi (x)

for some ai 2 (K [x℄> )[t℄, whi h is equivalent to g (x)hj (x) 2 hf1 ; : : : ; fk iK [x℄> for all j (set ti := 0 for all i, and then tj := 1 and ti = 0 for i 6= j ). Solution 2 : De ne h as above. We an ompute I1 : hhi by Lemma 1.8.12 and then I1 : I2 by eliminating t1 ; : : : ; tr 1 from I1 : hhi a

ording to Se tion 1.8.2. The same pro edure works with h := h1 + th2 + t2 h3 + + tr 1 hr 2 (K [x℄> )[t℄ with just one new variable t (Exer ise 1.8.2).


81

SINGULAR Example 1.8.13 (quotient of ideals). ring A=0,(x,y,z),dp; ideal I1=x,y; ideal I2=y2,z; quotient(I1,I2); //-> _[1℄=y _[2℄=x

//the built-in SINGULAR ommand

Now let us pro eed as des ribed in Lemma 1.8.12: ideal J1=interse t(I1,ideal(I2[1℄)); ideal J2=interse t(I1,ideal(I2[2℄)); J1; //-> J1[1℄=y2 J1/I2[1℄=1 implies I1:I2[1℄=A. J2; //-> J2[1℄=yz

J2[2℄=xz

J2/I2[2℄=<x,y> implies I1:I2[2℄=<x,y> and all together we obtain I1:I2=<x,y>: ideal K1=J1[1℄/I2[1℄; ideal K2=J2[1℄/I2[2℄,J2[2℄/I2[2℄; interse t(K1,K2); //-> _[1℄=y _[2℄=x

1.8.9 Saturation Let I1 ; I2 K [x℄> be as in Se tion 1.8.7. We onsider the quotient of I1 by powers of I2

I1 = I1 : I20 I1 : I21 I1 : I22 I1 : I23 : : : K [x℄> : Sin e K [x℄> is Noetherian, there exists an s su h that I1 : I2s = I1 : I2s+i for all i 0. Su h an s satis es

I1 : I21 :=

S

i0

I1 : I2i = I1 : I2s ;

and I1 : I2s is alled the saturation of I1 with respe t to I2 . The minimal su h s is alled the saturation exponent . If I1 is radi al, then the saturation exponent is 1.

Problem : Given ideals I1 ; I2 K [x℄> , we want to ompute generators for I1 : I21 and the saturation exponent. Solution : Set I (0) = I1 and ompute su

essively I (j+1) = I (j) : I2 , j 0, by any of the methods of Se tion 1.8.8. In ea h step he k whether I (j +1) I (j ) ,

82


by using Se tion 1.8.1. If s is the rst j when this happens, then I (s) = I1 : I21 and s is the saturation exponent.

Corre tness follows from I (j ) = I1 : I2j , whi h is a onsequen e of Lemma 1.8.14 (1). The above method is usually mu h faster than omputing I1 : I2j , sin e I2j an be ome quite large. To provide a geometri interpretation of ideal{quotient and saturation, we state the following:

Lemma 1.8.14. Let A be a ring and I1 ; I2 ; I3 ideals in A. (1) a) (I1 \ I2 ) : I3 = (I1 : I3 ) \ (I2 : I3 ), in parti ular I1 : I3 = (I1 \ I2 ) : I3 if I3 I2 , b) (I1 : I2 ) : I3 = I1 : (I2 I3 ). (2) If I1 is T prime and I2 6 I1 , then I1 : I2j = I1 for j 1. T (3) If I1 = ri=1 Ji with Ji prime, then I1 : I21 = I1 : I2 = I2 6Ji Ji . Proof. (1) is an easy exer ise. (2) I1 I1 : I2j is lear. Let gI2j 2 I1 . Sin e I2 6 I1 and I1 is radi al, j I2 6 I1 and we an nd an h 2 I2j su h that h 62 I1 and gh 2 I1 . Sin e I1 is prime, we have g 2 I1 . (3) follows from (1) and (2) sin e I2s 6 Ji if and only if I2 6 Ji : r T j =1

!

Ji

: I2s =

!

T

I2s 6Ji

(Ji

: I s) 2

\

T

I2s Ji

!

Ji

: Is 2

=

T

I2s 6Ji

Ji :

We shall see in Chapter 3 that inT a Noetherian ring ea h radi al ideal I1 has a prime de omposition I1 = ri=1 Ji with Ji prime. For the geometri interpretation of the ideal quotient and the saturation , we use the notations of Appendix A.2, respe tively A.3. We have

V (I1 ) =

r S i=1

V (Ji ) :

Moreover, we have I2 Ji if and only if V (Ji ) is a losed subs heme of V (I2 ). Hen e, the variety de ned by I1 : I2 is

V (I1 : I2 ) =

[

V (Ji )6V (I2 )

V (Ji ) :

In other words, if I1 is a radi al ideal, then V (I1 : I2 ) is the Zariski losure of V (I1 ) r V (I2 ). Note that V (h0i : I ) = supp(I ) := fP 2 Spe (A) j P AnnA (I )g, due to Lemma 2.1.41 below. More generally, for nitely generated ideals I1 ; I2 ,

V (I1 : I2 ) = supp (I2 + I1 )=I1

Spe (A=I1 ) :


83

Here is another example, where we do not know a priori whether we are dealing with a radi al ideal or not: given an ideal I K [x1 ; : : : ; xn ℄ and some point a = (a1 ; : : : ; an ) 2 V (I ) su h that V 0 := V (I ) r fag A n is Zariski

losed. We wish to know equations for V 0 , that is, some ideal I 0 su h that V 0 = V (I 0 ). At the moment, we only know that there exist su h ideal I 0 and an ideal J K [x℄ satisfying I = I 0 \ J and V (J ) = fag, but we neither know I 0 nor J . Now, sin e V (J ) = fag, some power of the maximal ideal ma = hx1 a1 ; : : : ; xn an i is ontained in J (see Lemma A.2.3). Now, sin e I 0 6 ma , it is not diÆ ult to show that I 0 : ma = I 0 , using the existen e of a primary de omposition (Theorem 4.1.4) and Exer ise 4.1.3. Hen e, again 0 1 0 using Exer ise 4.1.3, we an on lude that I : m1 a = I : ma = I , that is,

V (I ) r fag = V (I : m1 a ):

In general, however, a geometri interpretation of I1 : I2 is more diÆ ult and requires a areful study of the primary de ompositions of I1 and I2 . I1 : I2 , or even I1 : I21 , may not kill a whole omponent of V (I1 ), it may just redu e part of the stru ture. For example, if I1 = hxy 2 ; y 3 i, I2 = hx; y i, then

I1 : I2 = I1 : I21 = hxy2 ; y3 i : hx; yi = hy2 i ; hen e, V (I1 : I2 ) is set-theoreti ally the same as V (I1 ) (namely, the x{axis), just with a slightly redu ed stru ture (indi ated in Figure 1.4 by the small arrow pointing in y {dire tion).

V (I1 ) Fig. 1.4.

V (I1 : I2 )

Symboli pi tures of V (hxy 2 ; y 3 i) and V (hxy 2 ; y 3 i : hx; y i).

Saturation is an important tool in omputational proje tive geometry, f. Appendix A.5, in parti ular, Lemma A.5.2 and the subsequent dis ussion.

SINGULAR Example 1.8.15 (saturation). ring A =0,(x,y,z),dp; ideal I1=x5z3,xyz,yz4; ideal I2=z; LIB"elim.lib"; sat(I1,I2); //-> [1℄: //-> _[1℄=y //-> _[2℄=x5 //-> [2℄: //-> 4

//the SINGULAR pro edure //the result

//the saturation exponent

84

1. Rings, Ideals and Standard Bases ideal J=quotient(I1,I2); //the way des ribed above int k; while(size(redu e(J,std(I1)))!=0) { k++; I1=J; J=quotient(I1,I2); } J; //-> J[1℄=y J[2℄=x5 k; //-> 4 //we needed to take the quotient 4 times

1.8.10 Kernel of a Ring Map Let ' : R1 := (K [x℄>1 )=I ! (K [y ℄>2 )=J =: R2 be a ring map de ned by polynomials '(xi ) = fi 2 K [y ℄ = K [y1 ; : : : ; ym ℄ for i = 1; : : : ; n (and assume that the monomial orderings satisfy 1 >2 LM(fi ) if 1 >1 xi , f. Lemma 1.5.8). If J = hg1 ; : : : ; gs i, gi 2 K [y ℄ and I = hh1 ; : : : ; ht i, hi 2 K [x℄, then de ne J0 := hg1 ; : : : ; gs iK [y℄ and set I0 := hh1 ; : : : ; ht iK [x℄ . Then ' is indu ed by '~ : K [x℄=I0 ! K [y℄=J0, xi 7! fi , and we have a ommutative diagram

K [x℄=I0

'~ /

_

R1

K [y℄=J0 _

'

/

R2 :

Problem: Let I; J and ' be as above. Compute generators for Ker('). Solution: Set H := hh1 ; : : : ; ht ; g1 ; : : : ; gs ; x1 f1; : : : ; xn fni K [x; y℄, and

ompute H 0 := H \ K [x℄ by eliminating y1 ; : : : ; ym from H ( f. Se tion 1.8.2). Then H 0 generates Ker(') by the following lemma.

Lemma 1.8.16. With the above notations, Ker(') = Ker('~)R1 and Ker('~) = I0 + hg1 ; : : : ; gs ; x1

f1 ; : : : ; x n

fn iK [x;y℄ \ K [x℄ mod I0 :

In parti ular, if >1 is global, then Ker(') = Ker('~). Proof. Obviously Ker('~)R1 Ker('). On the other hand, let h 2 Ker('), where h = (h1 =h2 ) + I for some h1 2 K [x℄, h2 2 S>1 , then h1 + I0 2 Ker('~). We on lude that Ker(') = Ker('~)R1 . Now let h 2 K [x℄ satisfy '~(h + I0 ) = 0, in other words, there exist polynomials a1 ; : : : ; as 2 K [y ℄ su h that


h(f1 ; : : : ; fn) +

s X j =1

85

aj gj = 0 :

Applying Taylor's formula to the polynomial h(x), we obtain

h(x) = h(f1 ; : : : ; fn ) + +

n X i=1

h (f ; : : : ; fn ) xi xi 1

fi

n 1 X 2h (f ; : : : ; fn ) xi 2 i;j =1 xi xj 1

fi

xj

fj

+::: :

This implies that, for suitable bi 2 K [x; y ℄,

h(x) +

n X i=1

bi (x; y) xi

fi (y) +

s X j =1

aj (y)gj (y) = 0 :

This implies that h 2 hg1 ; : : : ; gs ; x1 f1 (y ); : : : ; xn fn (y )iK [x;y℄ \ K [x℄. Conversely, let h 2 I0 + hg1 ; : : : ; gs ; x1 f1 (y ); : : : ; xn fn (y )iK [x;y℄ \ K [x℄,

h = h1 +

s X i=1

ai gi +

n X i=1

bi (xi

fi ); h1 2 I0 :

Substituting xi by fi we obtain

h(f1 ; : : : ; fn ) = h1 (f1 ; : : : ; fn) +

s X i=1

ai (f1 ; : : : ; fn ; y)gi :

But h1 (f1 ; : : : ; fn ) 2 J0 and g1 ; : : : ; gs 2 J0 , hen e, h(f1 ; : : : ; fn ) 2 J0 , whi h proves the laim.

Remark 1.8.17. Given a ring map '~ : A ! B , and J B an ideal, then '~ indu es a ring map ' : A ! B=J and Ker(') = '~ 1 (J ). Hen e, the same method for omputing the kernel an be used to ompute preimages of ideals. Sin e Ker(') = ' 1 (0), to ompute kernels or preimages is equivalent. Singular has the built{in ommand preimage.

SINGULAR Example 1.8.18 (kernel of a ring map). ring A=0,(x,y,z),dp; ring B=0,(a,b),dp; map phi=A,a2,ab,b2; ideal zero; setring A; preimage(B,phi,zero);

// ompute the preimage of 0 //the built-in SINGULAR ommand

86

1. Rings, Ideals and Standard Bases //-> _[1℄=y2-xz ring C=0,(x,y,z,a,b), dp; //the method des ribed above ideal H=x-a2, y-ab, z-b2; eliminate(H,ab); //-> _[1℄=y2-xz

1.8.11 Algebrai Dependen e and Subalgebra Membership Re all that a sequen e of polynomials f1 ; : : : ; fk 2 K [x1 ; : : : ; xn ℄ is alled algebrai ally dependent if there exists a polynomial g 2 K [y1; : : : ; yk ℄ r f0g satisfying g (f1 ; : : : ; fk ) = 0. This is equivalent to Ker(') 6= 0, where ' : K [y1; : : : ; yk ℄ ! K [x1 ; : : : ; xn ℄ is de ned by '(yi ) = fi . Ker(') an be

omputed a

ording to Se tion 1.8.10, and any g 2 Ker(') r f0g de nes an algebrai relation between the f1 ; : : : ; fk . In parti ular, f1 ; : : : ; fk are algebrai ally independent if and only if Ker(') = 0 and this problem was solved in Se tion 1.8.10. Related, but slightly dierent is the subalgebra{membership problem.

Problem: Given f 2 K [x1 ; : : : ; xn ℄, we may ask whether f is an element of the subalgebra K [f1 ; : : : ; fk ℄ K [x1 ; : : : ; xn ℄ = K [x℄. Solution 1: De ne : K [y0 ; : : : ; yk ℄ ! K [x℄, y0 7! f , yi 7! fi , ompute

Ker( ) a

ording to Se tion 1.8.10 and he k whether Ker( ) ontains an element of the form y0 g (y1 ; : : : ; yk ). That is, we de ne an elimination ordering for x1 ; : : : ; xn on Mon(x1 ; : : : ; xn ; y0 ; : : : ; yk ) with y0 greater than y1 ; : : : ; yk (for example, (dp(n), dp(1), dp(k))) and ompute a standard basis G of hy0 f; y1 f1 ; : : : yk fk i. Then G ontains an element with leading monomial y0 if and only if f 2 K [f1 ; : : : ; fk ℄. Solution 2: Compute a standard basis of hy1 f1; : : : ; yk fk i for an elimination ordering for x1 ; : : : ; xn on Mon(x1 ; : : : ; xn ; y1 ; : : : ; yk ) and he k whether the normal form of f with respe t to this standard basis does not involve any xi . This is the ase if and only if f 2 K [f1 ; : : : ; fk ℄ and the normal form expresses f as a polynomial in f1 ; : : : ; fk . We omit the proofs for these statements ( f. Exer ise 1.8.10). Note that f 2 K [f1 ; : : : ; fk ℄ implies a relation h(f; f1 ; : : : ; fk ) = 0 with h(y0 ; y1 ; : : : ; yk ) = y0 g(y1 ; : : : ; yn ), hen e f; f1 ; : : : ; fk are algebrai ally dependent (the onverse does not need to be true). Note further that the map ' : K [y1 ; : : : ; yk ℄ ! K [x1 ; : : : ; xn ℄, yi ! fi (x) is surje tive if and only if xi 2 K [f1 ; : : : ; fk ℄ for all i. Hen e, Solution 1 or Solution 2 an be used to he k whether a given ring map is surje tive.


87

SINGULAR Example 1.8.19 (algebrai dependen e). ring A=0,(x,y),dp; poly f=x4-y4; poly f1=x2+y2; poly f2=x2-y2; LIB"algebra.lib"; algDependent(ideal(f,f1,f2))[1℄;//a SINGULAR pro edure //-> 1 ring B=0,(u,v,w),dp; setring A; ideal zero; map phi=B,f,f1,f2; setring B; preimage(A,phi,zero); //-> _[1℄=vw-u

//the method des ribed above

//the kernel of phi //f=f1*f2 and hen e f,f1,f2 //are algebrai ally dependent

SINGULAR Example 1.8.20 (subalgebra membership). ring A=0,(x,y),dp; poly f,f1,f2=x4-y4,x2+y2,x2-y2; LIB"algebra.lib"; inSubring(f,ideal(f1,f2));//a SINGULAR pro edure //-> [1℄: //-> 1 //means f is ontained in K[f1,f2℄ //-> [2℄: //-> y(1)*y(2)-y(0) //means f1*f2-f=0

Another

Singular

pro edure whi h also tests subalgebra membership is

algebra_ ontainment.

Now let us pro eed as explained in the text: ring B = 0,(x,y,u,v,w),(dp(2),dp(1),dp(2)); //solution 1 ideal H=u-imap(A,f),v-imap(A,f1),w-imap(A,f2); std(H); //-> _[1℄=u-vw _[2℄=2y2-v+w _[3℄=x2-y2-w

Sin e u appears as a leading monomial, f of u vw in H implies f = f1 f2 .

2 K [f1 ; f2℄. Moreover, the existen e

ring C=0,(x,y,v,w),(dp(2),dp(2)); //solution 2 ideal H=v-imap(A,f1), w-imap(A,f2); poly f=imap(A,f); redu e(f,std(H)); //-> vw //again we find f=f1*f2

88


Exer ises

1.8.1. Let I1 ; I2 be two ideals in K [x℄> with I2 = hh1 ; : : : ; hr i, hi 2 K [x℄. De ne h := h1 + th2 + t2 h3 + + tr 1 hr 2 K [x; t℄. Prove that I1 : I2 = (I1 : h) \ K [x℄> :

1.8.2. Let I := hx2 + 2y2 3; x2 + xy + y2 3i Q [x; y℄. Compute the interse tions I \ Q [x℄ and I \ Q [y ℄.

1.8.3. Let ' : Q 2 ! Q 4 be the map de ned by (s; t) 7! (s4 ; s3 t; st3 ; t4 ). Com-

pute the Zariski losure of the image, '(Q 2 ), and de ide whether '(Q 2 ) oin ides with its losure or not.

1.8.4. Compute all omplex solutions of the system x2 + 2y2 x2 + xy + y2

2=0 2 = 0:

1.8.5. Che k whether the polynomial x2 + 5x is in the radi al of the ideal I = hx2 + y3 ; y7 + x3 y5 iK [x;y℄, respe tively in IK [x; y℄hx;yi. 1.8.6. Let > be any monomial ordering on Mon(x1 ; : : : ; xn ), let I K [x℄> be an ideal, and let G be a standard basis of I with respe t to >. Show that the following are equivalent:

(1) dimK (K [x℄> =I ) < 1, (2) for ea h i = 1; : : : ; n there exists an ni 0 su h that xni i is a leading monomial of an element of G. (Hint: Use Exer ise 1.7.22)

1.8.7. Let K [x℄ be the polynomial ring in one variable, and let f 2 K [x℄ de-

ompose into linear fa tors, f = (x a1 )n1 : : : (x ar )nr for pairwise different ai 2 K . Show that K [x℄=hf i = K [x℄=hx a1 in1 K [x℄=hx ar inr and on lude that dimK K [x℄=hf i = n1 + + nr .

1.8.8. Let I = hf1 ; : : : ; fk i K [x1 ; : : : ; xn ℄ be an ideal. Use a lexi ographi al

Grobner basis of I to show that dimK (K [x℄=I ) < 1 if and only if then system of equations f1 = : : : = fk = 0 has only nitely many solutions in K , where K denotes the algebrai losure of K . (Hint: use indu tion on n, the previous exer ises and Appendix A.)

1.8.9. Prove statement (1) of Lemma 1.8.14. 1.8.10. Prove that Solutions 1 and 2 to the subalgebra{membership problem in Se tion 1.8.11 are orre t.

1.8.11. Use Singular to he k whether the line de ned by x + y = 3 (respe tively x + y = 500) and the ir le de ned by x2 + y 2 = 2 interse t.


89

1.8.12. Compute the kernel of the ring map Q [x; y; z ℄ ! Q [t℄=ht12 i de ned by x 7! t5 , y 7! t7 + t8 , z 7! t11 .

1.8.13. Show that the ring Q [s4 ; s3 t; st3 ; t4 ℄ is isomorphi to

Q [x1 ; x2 ; x3 ; x4 ℄=I with I = hx2 x3

x21 x3 ; x1 x23 x22 x4 i. 1.8.14. Create a homogeneous polynomial p of degree 3 in three variables with random oeÆ ients and use the lift ommand to express p as a linear

ombination of the partial derivatives of p. x1 x4 ; x33

x2 x24 ; x32

A. Geometri Ba kground

Es ist die Freude an der Gestalt in einem hoheren Sinne, die den Geometer ausma ht. Alfred Clebs h

In this appendix we introdu e a few on epts of algebrai geometry in order to illustrate some of the algebrai onstru tions used in this book. These illustrations are meant to stimulate the reader to develop his own geometri intuition on what is going on algebrai ally. Indeed, the onne tion between algebra and geometry has turned out to be very fruitful and both merged to be ome one of the leading areas in mathemati s: algebrai geometry. In order to provide some geometri understanding, we present, in this appendix, the very basi on epts of lassi al aÆne and proje tive varieties over an algebrai ally losed eld K (say C ). However, we also introdu e the modern ounterparts, Spe and Proj, whi h bridge, quite naturally and with less assumptions, the anyon between algebra and geometry. One word about the role of the ground eld K . Algebrai geometers usually draw real pi tures, think about them as omplex varieties and perform

omputations over some nite eld. We re ommend following this attitude, whi h is justi ed by su

essful pra ti e. Moreover, the modern language of s hemes even allows one to formulate and prove geometri statements for arbitrary elds whi h oin ide with the lassi al pi ture if the eld is algebrai ally losed ( f. A.5). For a deeper study of algebrai geometry, we re ommend the book of Hartshorne [97℄, whi h is not only the standard referen e book but also represents modern algebrai geometry in an ex ellent way, omplemented perhaps by the red book of Mumford [145℄, and the books of Harris [95℄ and Brieskorn and Knorrer [25℄.

A.1 Introdu tion by Pi tures The basi problem of algebrai geometry is to understand the set of solutions

x = (x1 ; : : : ; xn ) 2 K n of a system of polynomial equations

f1 (x1 ; : : : ; xn ) = 0 ; : : : ; fk (x1 ; : : : ; xn ) = 0 ;

404


2 K [x℄ = K [x1 ; : : : ; xn ℄ and K a eld. The solution set is alled an algebrai set or algebrai variety. The pi tures in Figures A.1 { A.8 show examples of algebrai varieties. However, algebrai sets really live in dierent worlds, depending on whether K is algebrai ally losed or not. For instan e, the question whether the simple polynomial equation xn + y n + z n = 0, n 3, has any non{trivial solution in K , is of fundamental dieren e if we ask this for K to be C , R or Q . (For C we obtain a surfa e, for R we obtain a surfa e if n is odd but only f0g if n is even, and for Q this is Fermat's problem, solved by A. Wiles in 1994.) Classi al algebrai geometry assumes K to be algebrai ally losed. Real algebrai geometry is a eld in its own and the study of varieties over Q belongs to arithmeti geometry, a merge of algebrai geometry and number theory. In this appendix we assume K to be algebrai ally losed. Many of the problems in algebra, in parti ular, omputer algebra, have a geometri origin. Therefore, we hoose an introdu tion by means of some pi tures of algebrai varieties, some of them being used to illustrate subsequent problems. The pi tures in this introdu tion, Figures A.1 { A.8, were not only hosen to illustrate the beauty of algebrai geometri obje ts but also be ause these varieties have had some prominent in uen e on the development of algebrai geometry and singularity theory. The Clebs h ubi itself has been the obje t of numerous investigations in global algebrai geometry, the Cayley and the D4 { ubi also, but, moreover, sin e the D4 { ubi deforms, via the Cayley ubi , to the Clebs h ubi , these rst three pi tures illustrate deformation theory, an important bran h of ( omputational) algebrai geometry. The ordinary node, also alled A1 {singularity (shown as a surfa e singularity), is the most simple singularity in any dimension. The Barth sexti illustrates a basi but very diÆ ult and still (in general) unsolved problem: to determine the maximum possible number of singularities on a proje tive variety of given degree. Whitney's umbrella was, at the beginning of strati ation theory, an important example for the two Whitney onditions. We use the umbrella in Chapter 3 to illustrate that the algebrai on ept of normalization may even lead to a parametrization of a singular variety, an ultimate goal in many

ontexts, espe ially for graphi al representations. In general, however, su h a parametrization is not possible, not even lo ally, if the variety has dimension larger than one. For urves, on the other hand, the normalization is always smooth and, hen e, provides, at least lo ally, a parametrization. Indeed, omputing the normalization of the ideal given by the impli it equations for the spa e urve in Figure A.8, we obtain the given parametrization. Conversely, the equations are derived from the parametrization by eliminating the variable t. Elimination of variables is perhaps the most important basi appli ation of Grobner bases.

fi

A.1 Introdu tion by Pi tures

405

The Clebs h Cubi This is the unique ubi surfa e whi h has S5, the symmetri group of ve letters, as symmetry group. It is named after its dis overer Alfred Clebs h and has the aÆne equation 81(x3 + y 3 + z 3 ) 189(x2 y + x2 z + xy 2 + xz 2 + y 2 z + yz 2 ) + 54xyz + 126(xy + xz + yz ) 9(x2 + y 2 + z 2 )

The Cayley Cubi There is a unique ubi surfa e whi h has four ordinary nodes (see Fig. A.5), usually alled the Cayley ubi after its dis overer, Arthur Cayley. It is a degeneration of the Clebs h ubi , has S4 as symmetry group, and the proje tive equation is

A ubi with a D4 {singularity. Degenerating the Cayley ubi we get a D4 {singularity. The aÆne equation is x(x2 y2 ) + z2 (1 + z) + 25 xy + 52 yz = 0 :

The Barth Sexti . The equation for this sexti was found by Wolf Barth. It has 65 ordinary nodes, the maximal possible number for a sexti . Its aÆne equation is (with

= 1+2 5 ) (8 +4)x2 y 2 z 2 4 (x4 y 2 + y 4 z 2 + x2 z 4 ) + 2 (x2 y 4 + y 2 z 4 + x4 z 2 )

Fig. A.1.

9(x + y + z ) + 1 = 0 :

Fig. A.3.

Fig. A.2.

z0 z1 z2 + z0 z1 z3 + z0 z2 z3 + z1 z2 z3 = 0 :

Fig. A.4.

p

2 + 1 2 2 2 (x + y + z 1)2 = 0 : 4

406


Fig. A.5.

An ordinary node.

An ordinary node is the most simple singularity. It has the equation x2 + y2 z2 = 0 :

Fig. A.7.

degree 11.

A 5{nodal plane urve of

Deforming an A10 {singularity (normal form: y 11 x2 = 0) we obtain a 5{nodal plane urve of degree 11: 32x2 2097152y 11 + 1441792y 9 360448y 7 + 39424y 5 1760y 3 + 22y 1 :

Fig. A.6.

Whitney's Umbrella.

The Whitney umbrella is named after Hassler Whitney who studied it in onne tion with the strati ation of analyti spa es. It has the aÆne equation x2 y z 2 = 0 :

Fig. A.8.

A spa e urve.

This spa e urve is given parametri ally by x = t4 ; y = t3 ; z = t2 ; or impli itly by x z2 = y2 z3 = 0 :

Finally, the 5{nodal plane urve illustrates the singularities of plane

urves, in parti ular, the deformation of a urve singularity into a nodal

urve. Moreover, this kind of deformations, with the maximal number of nodes, also play a prominent role in the lo al theory of singularities. For in-


407

stan e, from this real pi ture we an read o the interse tion form and, hen e, the monodromy of the singularity A10 by the beautiful theory of A'Campo and Gusein-Zade. By means of standard bases in lo al rings, there exists a

ompletely dierent, algebrai algorithm to ompute the monodromy [160℄. Singular an be used to draw real pi tures of plane urves or of surfa es in 3{spa e, that is, of hypersurfa es de ned by polynomials in two or three variables. For this, Singular alls the programme surf written by Stephan Endra, whi h is distributed together with Singular, but whi h an also be used as a stand{alone programme (unfortunately, up to now, surf runs only under Linux and Sun{Solaris). Drawing \ni e" real pi tures depends very mu h on the hosen equation, for example, the s aling of the variables, and the kind of proje tion hosen by the graphi s system. It is, therefore, re ommended to experiment with the on rete equations and with the parameters within surf. Here is the Singular input for drawing the Whitney umbrella and a surprise surfa e:

SINGULAR Example A.1.1 (surfa e plot). LIB "surf.lib"; ring r = 0,(x,y,z),dp; poly f = -z2+yx2; map phi = r,x,y,z-1/4x-2; plot(phi(f));

//the Whitney umbrella

f =(2*x^2+y^2+z^2-1)^3-(1/10)*x^2*z^3-y^2*z^3; // A surprise surfa e (equation due to Tore Norstrand) phi = r,1/6z,1/6x,1/6y; //res aling plot(phi(f));

Let us now dis uss some geometri problems for whi h the book des ribes algebrai algorithms, whi h are implemented in Singular. Re all rst the most basi , but also most important, appli ations of Grobner bases to algebrai onstru tions (Sturmfels alled these \Grobner basi s"). These an be found in Chapters 1, 2 and 5 of this book.

Ideal (respe tively module) membership problem, Interse tion with subrings (elimination of variables), Interse tion of ideals (respe tively submodules), Zariski losure of the image of a map, Solvability of polynomial equations, Solving polynomial equations, Radi al membership, Quotient of ideals, Saturation of ideals, Kernel of a module homomorphism,

408


Kernel of a ring homomorphism, Algebrai relations between polynomials, Hilbert polynomial of graded ideals and modules. The next questions and problems lead to algorithms whi h are slightly more (some of them mu h more) involved. They are, nevertheless, still very basi and quite natural. We should like to illustrate them by means of four simple examples, shown in the Figures A.9 and A.10, referred to as Examples (A) { (D). We re ommend redoing the omputations using Singular, the appropriate ommands an be found in the hapters we refer to.

(A) The Hypersurfa e V (x2 + y 3 Fig. A.9.

(C) The Spa e Curve V (xy;xz;yz). Fig. A.10.

t2 y2 ).

(B) The Variety V (xz;yz ).

Examples (A) and (B).

(D) The Set of Points V (y 4 xy3 xy;x3 y xy;x4

y2 , x2 ).

Examples (C) and (D).

Assume we are given an ideal I K [x℄ = K [x1 ; : : : ; xn ℄, by a nite set of generators f1 ; : : : ; fk 2 K [x℄. Consider the following questions and problems: (1) Is V (I ) irredu ible or may it be de omposed into several algebrai va-

rieties? If so, nd its irredu ible omponents. Algebrai ally this means


to ompute a primary de omposition of I or of

ompute the asso iated prime ideals of I.

409

p

I, the latter means to

Example (A) is irredu ible; Example (B) has two omponents (one of dimension 2 and one of dimension 1); Example (C) has three (one{dimensional), and Example (D) has nine (zero{dimensional) omponents. Primary de omposition is treated in Chapter 4.

p

(2) Is I a radi al ideal (that is, I = I )? If not, ompute its radi al

p

I.

In Examples (A) { (C) the ideal I is radi al, while in Example (D) we have p I = hy3 y; x3 xi, whi h is mu h simpler than I . In this example the

entral point orresponds to V (hx; y i2 ) whi h is a fat point , that is, it is a solution of I of multipli ity (= dimK K [x; y ℄=hx; y i2 ) larger than 1 (equal to 3). All other points have multipli ity 1, hen e, the total number of solutions ( ounted with multipli ity) is 11. This is a typi al example of the kind B. Bu hberger (respe tively W. Grobner) had in mind at the time of writing his thesis, [28℄. We show in Chapter 4, Se tion 4.5, how to ompute the radi al. In Corollary 5.3.17 we show how to ompute the dimension as a K {ve tor spa e, respe tively a K {basis, of K [x1 ; : : : ; xn ℄=I if the quotient is nite dimensional.

(3) A natural question to ask is, how independent are the generators f1 ; : : : ; fk of I? That is, we ask for all relations (r1 ; : : : ; rk ) 2 K [x℄k su h that k X i=1

ri fi = 0 :

These relations form a submodule of K [x℄k , whi h is alled the syzygy module of f1 ; : : : ; fk and is denoted by syz(I ).1 It is the kernel of the K [x℄{linear module homomorphism

K [x℄k

! K [x℄ ;

(r1 ; : : : ; rk ) 7

!

k X i=1

ri fi :

Syzygies are introdu ed and omputed in Chapter 2, Se tion 2.5. (4) More generally, we may ask for generators of the kernel of a K [x℄{linear map K [x℄r ! K [x℄s or, in other words, for solutions of a system of linear equations over K [x℄. A dire t geometri interpretation of syzygies is not so lear, but there are instan es where properties of syzygies have important geometri onsequen es, f. [159℄. To ompute the kernel of a module homomorphism, see Chapter 2, Se tion 2.8.7.

1 In general, the notion syz(I ) is a little misleading, be ause the syzygy module depends on the hosen system of generators for I , see Chapter 2, Remark 2.5.2.

410


In Example (A) syz(I ) = 0, in Example (B) syz(I ) = h( y; x)i K [x; y; z ℄2 , in Example (C) syz(I ) = h( z; y; 0); ( z; 0; x)i K [x; y; z ℄3 , and in Example (D) syz(I ) K [x; y; z ℄4 is generated by the ve tors (x; y; 0; 0), (0; 0; x; y ) and (0; x2 1; y 2 + 1; 0). (5) A more geometri question is the following. Let V (I 0 ) V (I ) be a subvariety. How an we des ribe V (I ) r V (I 0 ) ? Algebrai ally, this amounts

to nding generators for the ideal quotient 2 I : I 0 = ff 2 K [x℄ j fI 0 I g :

Geometri ally, V (I : I 0 ) is the smallest variety ontaining V (I ) r V (I 0 ) whi h is the (Zariski) losure of V (I ) r V (I 0 ), a proof of this statement is given in Chapter 1, Se tion 1.8.9.

In Examples (B), (C) we ompute the ideal quotients hxz; yz i : hx; y i = hz i and hxy; xz; yz i : hx; y i = hz; xy i, whi h give, in both ases, equations for the omplement of the z {axis fx = y = 0g. In Example (D) we obtain I : hx; yi2 = hy(y2 1); x(x2 1); (x2 1)(y2 1)i, the orresponding zero{ set being eight points, namely V (I ) without the entral point. See Chapter 1, Se tions 1.8.8 and 1.8.9 for further properties of ideal quotients and for methods on how to ompute them. (6) Geometri ally important is the proje tion of a variety V (I ) K n onto a linear subspa e K n r . Given generators f1 ; : : : ; fk of I K [x1 ; : : : ; xn ℄, we want to nd generators for the ( losure of the) image of V (I ) K n in K n r = fx j x1 = = xr = 0g. The losure of the image is de ned by the ideal I \ K [xr+1 ; : : : ; xn ℄, and nding generators for this interse tion

is known as eliminating x1 ; : : : ; xr from f1 ; : : : ; fk .

Proje ting the varieties of Examples (A) { (C) to the (x; y ){plane is, in the rst two ases, surje tive and in the third ase it gives the two oordinate axes in the (x; y ){plane. This orresponds to the fa t that the interse tion with K [x; y ℄ of the rst two ideals is h0i, while the third one is hxy i. Proje ting the nine points of Example (D) to the x{axis we obtain, by eliminating y , the polynomial x2 (x 1)(x + 1), des ribing the three image points. This example is dis ussed further in Example A.3.13. The geometri ba kground of elimination is dis ussed in detail in A.2 and A.3. The algorithmi and omputational aspe ts are presented in Chapter 1, Se tion 1.8.2. (7) Another problem is related to the Riemann singularity removable theorem, whi h states that a fun tion of a omplex manifold, whi h is holomorphi and bounded outside a subvariety of odimension 1, is a tually holomorphi everywhere. This statement is well{known for open subsets of C . In higher dimensions there exists a se ond singularity removable theorem, whi h states that a fun tion, whi h is holomorphi outside a 2 The same de nition applies if I;I are submodules of K [x℄k . 0


411

subvariety of odimension 2 (no assumption on boundedness), is holomorphi everywhere. For singular omplex algebrai varieties this is not true in general, but those for whi h the two removable theorems hold are alled normal . Moreover, ea h redu ed variety has a normalization, and there is a morphism with nite bres from the normalization to the variety, whi h is an isomorphism outside the singular lo us. Given a variety V (I ) K n, the problem is to nd a normal variety V (J ) K m and a polynomial map K m ! K n indu ing the normalization map V (J ) ! V (I ). It an be redu ed to irredu ible varieties (but

need not be, as shown in Chapter 3, Se tion 3.6), and then the equivalent algebrai problem is to nd the normalization of K [x1 ; : : : ; xn ℄=I, that is, the integral losure of K [x℄=I in the quotient eld of K [x℄=I, and to present this ring as an aÆne ring K [y1; : : : ; ym℄=J for some m and J.

For Examples (A) { (C) it an be shown that the normalization is smooth. In (B) and (C), it is a tually the disjoint union of the smooth omponents. The

orresponding rings are K [x1 ; x2 ℄; K [x1 ; x2 ℄ K [x3 ℄; K [x1 ℄ K [x2 ℄ K [x3 ℄. (Use Singular as in Chapter 3, Se tion 3.6.) The fourth example (D) has no normalization, as it is not redu ed.

A related problem is to nd, for a non{normal variety V , an ideal H su h that V (H ) is the non{normal lo us of V . The normalization algorithm is des ribed in Chapter 3, Se tion 3.6. There, we also present an algorithm to ompute the non{normal lo us. In the examples above, the non{normal lo us is equal to the singular lo us. (8) The signi an e of singularities appears not only in the normalization problem. The study of singularities is also alled lo al algebrai geometry and belongs to the basi tasks of algebrai geometry. Nowadays, singularity theory is a whole subje t on its own ( f. A.9 for a short introdu tion). A singularity of a variety is a point whi h has no neighbourhood in whi h the Ja obian matrix of the generators has onstant rank. In Example (A) the whole t{axis is singular, in the other three examples only the origin.

One task is to ompute generators for the ideal of the singular lo us, whi h is itself a variety. This is just done by omputing subdeterminants of the Ja obian matrix, if there are no omponents of dierent dimensions. In general, however, we need, additionally, to ompute either an equidimensional de omposition or annihilators of Ext groups. For how to ompute the singular lo us see Chapter 5, Se tion 5.7.

In Examples (A) { (D), the singular lo us is given by the ideals hx; y i, hx; y; z i,

hx; y; z i, hx; yi2 , respe tively.

412


(9) Studying a variety V (I ), I = hf1 ; : : : ; fk i, lo ally at a singular point, say the origin of K n , means studying the ideal IK [x℄hxi , generated by I in

the lo al ring

K [x℄hxi =

f f; g 2 K [x℄; g 62 hx1 ; : : : ; xn i : g

In this lo al ring the polynomials g with g(0) 6= 0 are units, and K [x℄ is a subring of K [x℄hxi . Now all the problems we onsidered above an be formulated for ideals in K [x℄hxi and modules over K [x℄hxi instead of K [x℄. The geometri problems should be interpreted as properties of the variety in a neighbourhood of the origin, or more generally, the given point. At rst glan e, it seems that omputation in the lo alization K [x℄hxi requires omputation with rational fun tions. It is an important fa t that this is not ne essary, but that basi ally the same algorithms whi h were developed for K [x℄ an be used for K [x℄hxi . This is a hieved by the hoi e of a spe ial ordering on the monomials of K [x℄ where, loosely speaking, the monomials of lower degree are onsidered to be larger. A systemati study is given in Chapter 1. In A.8 and A.9 we give a short a

ount of lo al properties of varieties and of singularities. All the above problems have algorithmi and omputational solutions, whi h use, at some pla e, Grobner basis methods. Moreover, algorithms for most of these have been implemented quite eÆ iently in several omputer algebra systems. Singular is also able to handle, in addition, lo al questions systemati ally.

A.2 AÆne algebrai varieties From now on, we always assume K to be an algebrai ally losed eld, ex ept when we spe ify K otherwise. We start with the simplest algebrai varieties, aÆne varieties.

De nition A.2.1. A n = A nK denotes the n{dimensional aÆne spa e over K , the set of all n{tuples x = (x1 ; : : : ; xn ) with xi 2 K , together with its

stru ture as aÆne spa e. A set X A nK is alled an aÆne algebrai set or a ( lassi al) aÆne algebrai variety or just an aÆne variety (over K ) if there exist polynomials f 2 K [x1 ; : : : ; xn ℄, in some index set , su h that

X = V (f )2 = fx 2 A nK j f (x) = 0; X is then alled the zero{set of (f )2 .

8 2 g :

A.2 AÆne algebrai varieties

413

If L is a non{algebrai ally losed eld and f are elements of L[x℄, we may

onsider an algebrai ally losed eld K ontaining L (for example, K = L, the algebrai losure of L) and all statements apply to the f onsidered as elements of K [x℄. Of ourse, X depends only on the ideal I generated by the f , that is, X = V (I ) with I = hf j 2 iK [x℄ . By the Hilbert basis theorem, 1.3.5, there are nitely many polynomials su h that I = hf1 ; : : : ; fk iK [x℄ and, hen e, X = V (f1 ; : : : ; fk ). If X = V (f1 ; : : : ; fk ) with deg(fi ) = 1, then X A n is an aÆne linear subspa e of A n . If X = V (f ) for a single polynomial f 2 K [x℄ r f0g then X is alled a hypersurfa e in A n . A hypersurfa e in A 2 is alled an (aÆne) plane

urve and a hypersurfa e in A 3 an aÆne surfa e in 3{spa e . Hypersurfa es in A n of degree 2, 3, 4, 5, : : : are alled quadri s, ubi s, quarti s, quinti s, :::. Figure A.11 shows pi tures of examples with respe tive equations. These

an be drawn using Singular as indi ated in the following example:

SINGULAR Example A.2.2 (surfa e plot). ring r=0,(x,y,z), dp; poly f= ...; LIB"surf.lib"; plot(f);

The pi tures shown in Figure A.11 give the orre t impression that the varieties be ome more ompli ated if we in rease the degree of the de ning polynomial. However, the pi tures are real, and it is quite instru tive to see how they hange if we hange the oeÆ ients of terms (in parti ular the signs). The quinti in Figure A.11 is the Togliatti quinti , whi h embellishes the over of this book.

Lemma A.2.3. For ideals I; Ii ; I K [x1 ; : : : ; xn ℄, any index set, we have

(1) ; = V (h1i), A nK = V (h0i); Q S T (2) ki=1 V (Ii ) = V ki=1 Ii = V ki=1 Ii ; P S T (3) 2 V (I ) = V 2 I = V 2 I ; (4) V (I1 ) V (pI2 ) if I1 I2 ; (5) V (I ) = V I ; p p (6) V (I1 ) = V (I2 ) if and only if I1 = I2 .

The easy proof is left to the reader, (6) being a onsequen e of Hilbert's Nullstellensatz (Theorem 3.5.2). A dire t onsequen e is

Lemma A.2.4. (1) ;, A n are aÆne varieties. (2) The union of nitely many aÆne varieties is aÆne.

414


x2 y 2 z 2 = 0

x4 + y4 + z4 + 1 x2 y2 z 2 y 2 z 2 z 2 x2 x 2 y 2 = 0

Fig. A.11.

x3 y 2 z 2 = 0

64 (x 1) x4 4x3 10x2 y 2 4x2 +16xp 20xy 2 +5y 4 + p 16 20y 2 p p 5 5 5 5 2z p5 4 (x2 + y2 z2 ) + 1 + 3 52 = 0

Quadri , ubi , quarti , quinti hypersurfa es.

(3) The interse tion of arbitrary many aÆne varieties is aÆne. The lemma says that the aÆne algebrai sets in A n are the losed sets of a topology. This topology is alled the Zariski topology on A n . For an algebrai set X A n , the indu ed topology is alled the Zariski topology on X . Hen e, an aÆne algebrai set is the same as a (Zariski{) losed (that is, losed in the Zariski topology) subset of an aÆne spa e. If Y X is losed, we all Y also a ( losed) subvariety of X . X A n is

alled quasi{aÆne if it is lo ally losed , that is, the interse tion of an open and a losed subset. Note that we an identify, as a set, A n with A 1 A 1 , but the Zariski topology on A n is not the produ t of the Zariski topologies of A 1 (for example, the only non{trivial losed sets in the produ t topology of A 1 A 1 are nite unions of points and lines). Having de ned the zero{set of an ideal, we de ne now the ideal of a set.


De nition A.2.5. For any set X A n de ne

415

2 K [x1 ; : : : ; xn ℄ f jX = 0 ; the (full vanishing) ideal of X , where f jX : X ! K denotes the polynomial I (X ) := f

fun tion of f restri ted to X .

Lemma A.2.6. Let X A n be a subset, X1 ; X2 A n aÆne varieties. (1) (2) (3) (4) (5)

I (X ) is a radi al ideal. V I (X ) = X the Zariski losure of Xin A n . If X is an aÆne variety, then V I (X ) = X. I (X ) = I (X ). X1 X2 if and only if I (X2 ) I (X1 ), X1 = X2 if and only if I (X1 ) = I (X2 ). (6) I (X1 [ X2 ) = Ip(X1 ) \ I (X2 ). (7) I (X1 \ X2 ) = I (X1 ) + I (X2 ). The proof is left as an exer ise. It follows that, for a losed set X , the ideal I (X ) determines the aÆne algebrai set X (and vi e versa), showing already a tight onne tion between ideals of K [x℄ and aÆne algebrai sets in A nK . However, I (X ) is abstra tly de ned, we only know (by the Hilbert basis theorem) that it is nitely generated, but, given X , we do not know a set of generators of I (X ). Therefore, given any ideal I su h that X = V (I ), we may ask how far does I dier from I (X p) or, how far an we re over I from its zero{set V (I )? Of ourse, V (I ) = V ( I ), that is, we an re over I at most up to radi al. Hilbert's Nullstellensatz (Theorem 3.5.2) says that, for algebrai ally losed elds, this is the only ambiguity. That is, if I K [x1 ; : : : ; xn ℄ is an ideal, K is algebrai ally losed, and X = V (I ), then

p

p

I (X ) = I :

Note that the in lusion I I (X ) holds for any eld. The other in lusion does not hold for K not algebrai ally losed. Consider, p for example, K = R , I := hx2 + y2 + 1i R [x; y℄. Then V (I ) = ;, but I = I ( R [x; y℄ = I (;). As a onsequen e, we obtain, for K algebrai ally losed, an in lusion reversing bije tion (HN refers to Hilbert's Nullstellensatz)

faÆne algebrai sets in A nK g o HN / fradi al ideals I K [x1 ; : : : ; xn ℄g X V (I ) o

/ I (X )

I:

416


Corollary A.2.7. Let K [x℄ = K [x1 ; : : : ; xn ℄. Then

(1) V (I ) 6= ; for any proper ideal I ( K [x℄. (2) If m K [x℄ is a maximal ideal then

m = hx1

p1 ; : : : ; x n

pn iK [x℄

for some point p = (p1 ; : : : ; pn ) 2 A n . In parti ular, V (m) = fpg.

p

Proof. (1) X = V (I ) = ; implies I (X ) = K [x℄ and, hen e, I = K [x℄ by the Nullstellensatz, whi h implies I = K [x℄. (2) By (1) there exists some point p = (p1 ; : : : ; pn ) 2 V (m). The orresponding ideal Ip := hx1 p1 ; : : : ; xn pn i K [x℄ is maximal and satis es V (Ip ) = fpg V (m). Hen e, m Ip and, therefore, m = Ip , sin e m is max-

imal.

We see, in parti ular, that the bije tion HN indu es a bije tion between points in A nK and maximal ideals in K [x℄. (This will be the link between lassi al aÆne varieties and aÆne s hemes de ned in A.3.) Moreover, we all an aÆne algebrai set X A nK irredu ible if X 6= ;, and if it is not the union of two proper aÆne algebrai subsets. With this de nition, the irredu ible algebrai sets in A nK orrespond to prime ideals in K [x℄ ( f. Proposition 3.3.5). Hen e, for K an algebrai ally losed eld, we have the following in lusion reversing bije tions (with K [x℄ = K [x1 ; : : : ; xn ℄):

faÆne algebrai setsSin A nK g HN! fradi alSideals in K [x℄g firredu ible aÆne algebrai setsSin A nK g ! fprimeSideals in K [x℄g fpoints of A nK g ! fmaximal ideals in K [x℄g : In parti ular, the irredu ible omponents of an aÆne algebrai set V (I ) are pre isely V (P1 ); : : : ; V (Pr ), where P1 ; : : : ; Pr are the minimal asso iated primes of I , that is, V (I ) = V (P1 ) [ [ V (Pr ) is the unique de omposition of V (I ) into irredu ible aÆne algebrai sets, no one ontaining another.

De nition A.2.8. Let X be an aÆne algebrai set, then the aÆne ring K [X ℄ := K [x1 ; : : : ; xn ℄=I (X ) is alled the oordinate ring of X , and the elements of K [X ℄ are alled regular fun tions on X . We de ne the dimension of an aÆne algebrai set X to be the dimension of its oordinate ring K [X ℄. In parti ular, if X1 ; : : : ; Xr are the irredu ible

omponents of X , then dim(X ) = max dim(X1 ); : : : ; dim(Xr ) , by Lemma 3.3.9.


417

De nition A.2.9. Let X A nK , Y A m K be aÆne algebrai sets. A map

f : X ! Y is alled a morphism of algebrai sets if there exists a polynomial e e e map fe = (fe1 ; : : : ; fem ) : A nK ! A m K , fi 2 K [x1 ; : : : ; xn ℄, su h that f = f jX . f is alled a (polynomial) representative of f . Mor(X; Y ) denotes the set of all morphisms from X to Y . An isomorphism is a bije tive morphism with f 1 also a morphism. It is easy to see that the omposition of morphisms is again a morphism. Moreover, morphisms are ontinuous in the Zariski topology: if Z Y is

losed with I (Z ) = hg1 ; : : : ; gk iK [Y ℄ , then f 1 (Z ) = fx 2 X j gi Æ fe(x) = 0g is losed, too. If fe = (fe1 ; : : : ; fem ), ge = (ge1 ; : : : ; gem ) : A nK ! A m K are two polynomial representatives of f then (fei gei )(x) = 0 for all x 2 X , hen e, fei gei 2 I (X ). In parti ular, we obtain a bije tion Mor(X; A 1K ) = K [X ℄ :

More generally, Mor(X; Y ) = fe 2 K [x1 ; : : : ; xn ℄m fe(X ) Y mod I (X ). Sin e any algebrai ally losed eld is in nite, f jA nK = 0 implies f = 0, hen e the oordinate ring of A nK is K [x℄ := K [x1 ; : : : ; xn ℄. By the above remarks, we have m Mor(X; A m K ) = K [X ℄

A mK , Mor(X; Y ) = f 2 K [X ℄m f (X ) Y :

and, more generally, for any losed subvariety Y

In the following we point out that there is a tight relation between morphisms

X ! Y and K -algebra homomorphisms K [Y ℄ ! K [X ℄: let f : X ! Y be a morphism with polynomial representative fe = (fe1 ; : : : ; fem ) : A nK ! A m K. Then, for g 2 K [y ℄ := K [y1 ; : : : ; ym ℄, we set f (g) := [g Æ fe℄ ;

the lass of g fe1 ; : : : ; fem in K [x1 ; : : : ; xn ℄=I (X ), whi h is independent of the

hosen representative fe. The latter de nes a map f : K [y ℄ ! K [x℄=I (X ), whi h is easily he ked to be a K {algebra homomorphism. Moreover, if g 2 I (Y ), then g Æ fejX = 0 and, hen e, f I (Y ) I (X ). Altogether, we see that a morphism f : X ! Y between algebrai sets X A nK and Y A m K indu es a K {algebra homomorphism

f : K [Y ℄ = K [y℄=I (Y ) ! K [x℄=I (X ) = K [X ℄

of the orresponding oordinate rings (in the opposite dire tion). Conversely, let ' : K [Y ℄ = K [y ℄=I (Y ) ! K [x℄=I (X ) = K [X ℄ be a K {algebra homomorphism. Then we may hoose any representatives fei 2 K [x℄ of '([yi ℄) 2 K [x℄=I (X ), i = 1; : : : ; m, and de ne the polynomial map

418


fe := (fe1 ; : : : ; fem) : A nK

! A mK :

Sin e the possible hoi es for fei dier only by elements of I (X ), the polynomial fun tion fejX : X ! A m K is, indeed independent of the hosen representatives. Moreover, its image is ontained in Y : onsider the K {algebra homomorphism 'e : K[y ℄ ! K [x℄, de ned by 'e(yi ) := fei , whi h is a lift of ' and satis es 'e I (Y ) I (X ). It follows that, for ea h x 2 X and ea h g 2 I (Y ), we have g fe1 (x); : : : ; fem (x) = 'e(g ) (x) = 0, showing that fe(X ) Y . Altogether, we see that a K {algebra homomorphism ' : K [Y ℄ ! K [X ℄ indu es a morphism of algebrai sets

'# := fejX : X ! Y : As an example, onsider f : A 1 ! X = V (y 2 x3 ) A 2 , t 7! (t2 ; t3 ). The indu ed ring map is ' = f : K [x; y ℄=hy 2 x3 i ! K [t℄, indu ed by the K { algebra homomorphism 'e : K [x; y ℄ ! K [t℄, x 7! t2 , y 7! t3 , that is, f (g ) = g(t2 ; t3 ). ' indu es '# : A 1 ! X , t 7! 'e(x); 'e(y) = (t2 ; t3 ). Altogether, we see that (f )# = f and ('# ) = '. The following proposition shows that this is a general fa t. In Singular, morphisms between aÆne varieties have to be represented by the orresponding ring maps, see Chapter 1, in parti ular Se tions 1.1, 1.3 and 1.5 for de nitions and examples. p Proposition A.2.10. Let X A nK , Y A m K , Z A K be aÆne algebrai

sets and K [X ℄, K [Y ℄, K [Z ℄ the orresponding oordinate rings. (1) (idX ) = idK [X ℄, (idK [X ℄ )# = idX .

(2) For morphisms X

f g Z of algebrai sets we have ! Y ! (g Æ f ) = f Æ g : K [Z ℄ ! K [X ℄ ;

' ! K [Y ℄ ! K [X ℄ we have Æ ')# = '# Æ # : X ! Z :

and for K{algebra homomorphisms K [Z ℄ (

(3) (f )# = f for f : X ! Y , a morphism of algebrai sets; ('# ) = ' for ' : K [Y ℄ ! K [X ℄, a K{algebra homomorphism. All the statements are easy to he k and left as an exer ise.

Corollary A.2.11. A morphism f : X ! Y of aÆne varieties is an isomorphism if and only if f : K [Y ℄ ! K [X ℄ is an isomorphism of K{algebras.

In the language of ategories and fun tors, Proposition A.2.10 says that asso iating

X7

! K [X ℄;

(f : X

! Y ) 7 ! (f : K [Y ℄ ! K [X ℄)


419

de nes a ( ontravariant) fun tor from the ategory of aÆne algebrai varieties to the ategory of aÆne K {algebras. Indeed, this fun tor is an equivalen e onto the full sub ategory of all redu ed aÆne K {algebras (by the Hilbert Nullstellensatz). The following proposition gives a geometri interpretation of inje tive, respe tively surje tive, K {algebra homomorphisms.

Proposition A.2.12. Let f : X ! Y be a morphism of aÆne varieties and

f : K [Y ℄ ! K [X ℄ the orresponding map of oordinate rings. Then (1) f is surje tive if and only if the image f (X ) Y is a losed subvariety, and f : X ! f (X ) is an isomorphism. (2) f is inje tive if and only if f (X ) is dense in Y , that is, f (X ) = Y .

We all f : X ! Y a losed embedding or losed immersion if f (X ) Y is

losed and f : X ! f (X ) is an isomorphism; f is alled dominant if f (X ) is dense in Y . For the proof of Proposition A.2.12 we need an additional lemma. Moreover, we shall use that under the identi ation Mor(X; A 1K ) = K [X ℄ a morphism g : X ! A 1K satis es g (x) = 0 for all x 2 X if and only if g = 0 in K [X ℄. If Y X A n is any subset then we write

I (Y ) := IA n (Y ) := f IX (Y ) := f

2 K [x1 ; : : : ; xn ℄ f jY 2 K [X ℄ f jY = 0 :

=0 ;

Lemma A.2.13. Let f : X ! Y be a morphism of aÆne algebrai sets, then

IY f (X ) = IY f (X ) = Ker(f : K [Y ℄ ! K [X ℄) :

Proof. For g 2 K [Y ℄ we have g f (X) = f0g if and only if g f (X ) = f0g, sin e g is ontinuous. Sin e g f (X ) = f (g )(X ) = f0g if and only if f (g ) is the zero morphism, we obtain IY f (X ) = IY f (X ) = Ker(f ). Proof of Proposition A.2.12. (1) Let p 2 f (X ) and mp = IY (fpg) K [Y ℄ the maximal ideal of p, then mp IY f (X ) = Ker(f ), the latter equality being given by Lemma A.2.13. If f is surje tive, then the indu ed map f : K [Y ℄= Ker(f ) ! K [X ℄ is an isomorphism, and the same holds for f : K [Y ℄=mp =! K [X ℄=f (mp ) : Hen e, f (mp ) is a maximal ideal and orresponds, by the Hilbert Nullstellensatz, to a unique point, V f (mp ) = fq g. If X A n , Y A m , p = (p1 ; : : : ; pm ), and if f has the polynomial representative fe = (fe1 ; : : : ; fem ) 2 K [x℄m then mp = hy1 p1 ; : : : ; ym pm iK [Y ℄ ,

420


f (mp ) = hfe1 p1; : : : ; fem pmiK [X ℄ . Hen e, we obtain fqg = V f (mp ) = fe 1 (p) \ X = f 1 (p); and it follows that f (q) = p 2 f (X ) and that f is inje tive. We on lude that f (X ) is losed and that f : X ! f (X ) is bije tive. Finally, sin e f : K [f (X )℄ = K [Y ℄=IY f (X ) ! K [X ℄ is an isomorphism, f

is an isomorphism, too, by Corollary A.2.11. Conversely, if f (X ) is losed and if f : X ! f (X ) is an isomorphism then f : K [Y ℄=IY f (X ) ! K [X ℄ is an isomorphism by Corollary A.2.11. Hen e, f : K [Y ℄ ! K [X ℄ is surje tive. (2) Using Lemma A.2.13, we have Ker(f ) = 0 if and only if IY f (X ) = 0 whi h is equivalent to f (X ) = Y . f

For example, the proje tion A 2 X = V (xy 1) ! A 1 , (x; y ) 7! x, has the image f (X ) = A 1 r f0g. For g 2 K [A 1 ℄ = K [x℄ we have f (g )(x) = g (x), and we see that f is not surje tive ([y ℄ 2 K [x; y ℄=hxy 1i r Im(f )), hen e, f is not a losed embedding. But f is inje tive, as it should be, sin e f (X ) is dense in A 1 . The library algebra.lib ontains pro edures to test inje tivity, surje tivity and isomorphy of ring maps:

SINGULAR Example A.2.14 (inje tive, surje tive). LIB "algebra.lib"; ring R = 0,(x,y,z),dp; qring Q = std(z-x2+y3); ring S = 0,(a,b, ,d),dp; map psi = R,a,a+b, -a2+d3; is_inje tive(psi,R); //-> 1 is_surje tive(psi,R); //-> 0

// quotient ring R/ // a map from R to S, // x->a, y->a+b, z-> -a2+d3 // psi is inje tive // psi is not surje tive

qring T = std(ideal(d, -a2+b3));// // map hi = Q,a,b,a2-b3; // // // is_bije tive( hi,Q); //-> 1 //

quotient ring S/ map Q --> T between two quotient rings, x->a, y->b, z->a2-b3

hi is an isomorphism

Remark A.2.15. The reader might wonder whether there is an algebrai hara terization of f : X ! Y being surje tive. Of ourse, f has to be inje tive but the problem is to de ide whether f (X ) is losed in Y . In general, this is diÆ ult and a simple algebrai answer does not exist.


421

By a theorem of Chevalley ( f. [97℄, [95℄), f (X ) is onstru tible , that is, a nite union of lo ally losed subsets of Y . However, if f is a proje tive morphism of proje tive varieties, then f (X ) is losed. We prove this fa t in Se tion A.7, where we also explain the origin of the points in f (X ) r f (X ) ( f. Remark A.7.14). For the further study of morphisms, we need to onsider produ ts of varieties. This will be espe ially useful for an algorithmi treatment and, hen e, for

omputational aspe ts of images of aÆne varieties. First we identify A n A m with A n+m . In parti ular, we have the Zariski topology of A n+m on A n A m (whi h is not the produ t of the two Zariski topologies). If X = V (I ) A n and Y = V (J ) A m are aÆne algebrai sets de ned by ideals I K [x1 ; : : : ; xn ℄ and J K [y1 ; : : : ; ym ℄, then X Y A n+m is an aÆne algebrai set, sin e, as an easily be seen,

X Y = V (hI; J iK [x;y℄) :

De nition A.2.16. For a morphism f : X ! Y , we de ne the graph of f ,

f := (x; y )

2 X Y y = f (x)

:

If fe = (fe1 ; : : : ; fem ) 2 K [x℄m is a polynomial representative of f , then

Hen e,

2 X Y yi fei (x) = 0; i = 1; : : : ; m ; = (x; y ) 2 X A m yi fei (x) = 0; i = 1; : : : ; m : n+m is an aÆne algebrai set, f A f = (x; y )

f =V

I; y1 fe1; : : : ; ym fem

K [x;y℄

:

Remark A.2.17. The proje tions pr1 : X Y ! X and pr2 : X Y ! Y are indu ed by the in lusions j1 : K [x℄ ,! K [x; y ℄ and j2 : K [y ℄ ,! K [x; y ℄. For an ideal I K [x℄ with V (I ) = X , and a morphism f : X ! Y with polynomial representative (fe1 ; : : : ; fem ), j1 indu es an isomorphism K [x℄=I

=

! K [x; y℄=hI; y1

fe1; : : : ; ym femi ;

the inverse morphism being indu ed by K [x; y ℄ ! K [x℄, xi 7! xi , yj 7! fej . Ba k to geometry, we see that the proje tion pr1 : X Y ! X indu es an isomorphism 1 : f ! X of aÆne varieties: 1 is the restri tion of the proje tion A n A m ! A n , hen e 1 is a morphism, and the polynomial map A n ! A n A m , x 7! x; fe(x) , indu es an inverse to 1 . We obtain the following ommutative diagram

422


A n X Jo JJ

=

/X f 1 JJ yy JJ 2 yyyy J pr y 2 f JJJ $ |yy m Y

Y A n+m

A

where 2 is the restri tion of the se ond proje tion pr2 : X Y ! Y . It follows that any morphism f of aÆne varieties an be represented as a omposition of an in lusion and a proje tion. The following lemma ontains the geometri meaning of elimination . In parti ular, part (2) says that we an ompute an ideal de ning the losure of f (X ) by eliminating variables from an appropriate ideal J .

Lemma A.2.18. (1) Let f : X ! Y be a morphism of aÆne algebrai sets, then

IY f (X ) = IX Y ( f ) \ K [Y ℄ :

(2) Let X A n , Y A m be aÆne algebrai sets, and let IX K [x1 ; : : : ; xn ℄ be any ideal with X = V (IX ). Moreover, let f : X ! Y be a morphism, indu ed by fe = (fe1 ; : : : ; fem ) : A n ! A m with fei 2 K [x1 ; : : : ; xn ℄. De ne J := hIX ; y1 fe1; : : : ; ym femiK [x;y℄ : Then the losure of the image of f is given by f (X ) = V (J \ K [y1; : : : ; ym ℄ ) : Moreover, for IX = I (X ) the full vanishing ideal, we obtain IA n+m ( f ) = hI (X ); y1 fe1; : : : ; ym femiK [x;y℄ ; and the full vanishing ideal of f (X ) A m is

IA m f (X ) = IA n+m ( f ) \ K [y1 ; : : : ; ym ℄ :

Proof. (1) Note that Lemma A.2.13 gives IY f (X ) = Ker(f ) = Ker(2 ), sin e f = (1 ) 1 Æ 2 and sin e 1 is an isomorphism. Now the result follows, sin e we may onsider K [Y ℄ = K [y ℄=I (Y ) as a subalgebra of K [X Y ℄ = K [x; y ℄=hI (X ); I (Y )iK [x;y℄ and sin e 2 is the anoni al map K [Y ℄ ! K [X Y ℄=IX Y ( f ). For (2) note that K [x℄=IX ! K [x; y ℄=J is an isomorphism with inverse indu ed by xi 7! xi , yi 7! fej . Therefore, J is radi al if and only if IX is radi al, and then J \ K [y ℄ is apradi al ideal, too. Sin e V (J ) = f , we have J =pIA n+m ( f ) K [x; y ℄ by Hilbert's Nullp stellensatz. Therefore, J \ K [y ℄ = J \ K [y ℄ = IA n+m ( f ) \ K [y ℄, and (2)

follows from (1).

A.3 Spe trum and AÆne S hemes


423

Abstra t algebrai geometry, as introdu ed by Grothendie k, is a far rea hing generalization of lassi al algebrai geometry. One of the main points is that it allows the appli ation of geometri methods to arbitrary ommutative rings, for example, to the ring Z. Thus, geometri methods an be applied to number theory, reating a new dis ipline alled arithmeti geometry. However, even for problems in lassi al algebrai geometry, the abstra t approa h has turned out to be very important. For example, for polynomial rings over an algebrai ally losed eld, aÆne s hemes provide more stru ture than lassi al algebrai sets. In a systemati manner, the abstra t approa h allows nilpotent elements in the oordinate ring. This has the advantage of understanding and des ribing mu h better \dynami aspe ts" of a variety, sin e nilpotent elements o

ur naturally in the bre of a morphism, that is, when a variety varies in an algebrai family. The abstra t approa h to algebrai geometry has, however, the disadvantage that it is often far away from intuition, although a geometri language is used. A s heme has many more points than a lassi al variety, even a lot of non{ losed points. This fa t, although against any \ lassi al" geometri feeling, has, on the other hand, the ee t that the underlying topologi al spa e of a s heme arries more information. For example, the abstra t Nullstellensatz, whi h is formally the same as Hilbert's Nullstellensatz, holds without any assumption. However, sin e the geometri assumptions are mu h stronger than in the lassi al situation (we make assumptions on all prime ideals ontaining an ideal, not only on the maximal ideals), the abstra t Nullstellensatz is more a remark than a theorem and Hilbert's Nullstellensatz is not a onsequen e of the abstra t one. Nevertheless, the formal oin iden e makes the formulation of geometri results in the language of s hemes mu h smoother, and the relation between algebra and geometry is, even for arbitrary rings, as lose as it is for lassi al algebrai sets de ned by polynomials over an algebrai ally

losed eld. At the end of Se tion A.5, we shall show how results about algebrai sets

an, indeed, be dedu ed from results about s hemes (in a fun torial manner). In the following we assume, as usual, all rings to be ommutative with 1.

De nition A.3.1. Let A be a ring. Then Spe (A) := fP

A j P is a prime ideal g

is alled the (prime ) spe trum of A, and

Max(A) := fm A j m is a maximal ideal g

is alled the maximal spe trum of A. For X = Spe (A) and I

V (I ) := fP

2 X j P Ig

A an ideal

is alled the zero{set of I in X . Note that V (I ) = supp(A=I ).

424


As for lassi al aÆne varieties, we have the following relations (re all that prime ideals are proper ideals): (1) (2) (3) (4) (5) (6)

V (h1i) = ;, V (h0i) = X ; Sk T V (Ii ) = V ki=1 Ii ; i =1 P S T 2 I ; 2 I = V 2 V (I ) = V if I1 I2 then p V (I1 ) V (I2 ); V (I ) = V ( I ); p p V (I1 ) = V (I2 ) if and only if I1 = I2 .

Only the statements (2) and (6) are non{trivial. (2) follows, sin e, by de niT T tion, P 2 V ( ki=1 Ii ) if and only if P ki=1 Ii , whi h means due to Lemma S 1.3.12 that P Ii for some i, whi h is equivalent to P 2 ki=1 V (Ii ). (6) will follow from Lemma A.3.3 and Theorem A.3.4 below. Using the above properties (1) { (3) we an de ne on X the Zariski topology by de ning the losed sets of X to be the sets V (I ) for I A an ideal. Note that this is formally the same de nition as for lassi al varieties. Max(A) has the indu ed topology from Spe (A).

De nition A.3.2. Let X = Spe (A) and Y X any subset. The ideal I (Y ) :=

\

P 2Y

P

is alled the (vanishing) ideal of Y in X . As in the lassi al ase, we have the following

Lemma A.3.3. Let Y X = Spe (A) be a subset. (1) (2) (3) (4)

I (Y ) is a radi al ideal. V I (Y ) is the Zariski losure of Y in X. If Y is losed, then V I (Y ) = Y . Let Y denote the Zariski losure of Y in X, then I Y = I (Y ).

Proof. (1) If, for some a 2 A, the n{th power an is in the interse tion of prime ideals, then also a is in the interse tion. (2) V I (Y ) is losed and ontains Y . If W = V (J ) is losed in X and

ontains Y then, forTea h a 2 J , we have a 2 P for all P J , that is, for all J I (Y ) and, therefore, P 2 W .Hen e, a 2 P 2Y P = I (Y ). This implies V I (Y ) V (J ) = W , showing that V I (Y ) is the smallest losed subset of X ontaining Y . Finally, (3) and (4) are onsequen es of (2).

The following analogue of Hilbert's Nullstellensatz is sometimes alled the abstra t Nullstellensatz , whi h holds for Spe (A), A any ring.


425

Theorem A.3.4. Let X = Spe (A), and I A an ideal. Then

p

I V (I ) = I : Proof. The statement follows, sin e

I V (I ) =

\

P 2V (I )

P=

\

P 2Spe (A) P I

p

P = I;

where the last equality follows from Exer ise 3.3.1. Note that a point P 2 Spe (A) (or, more pre isely, the set fP g Spe (A)) need not be losed. Indeed, by Lemma A.3.3 and Theorem A.3.4 we have

fP g = V (P ) = fQ 2 Spe (A) j Q P g ; I (fP g) = I fP g where fP g denotes the Zariski losure of fP g in Spe (A).

=P;

Sin e P is always ontained in some maximal ideal, it follows that fP g is

losed in Spe (A) if and only if P is a maximal ideal. Hen e, Max(A) = fP

2 Spe (A) j P

is a losed pointg :

Re all that a topologi al spa e X is alled irredu ible if X 6= ; and whenever X = A1 [ A2 , with A1 ; A2 X losed, then X = A1 or X = A2 . X is alled redu ible if it is not irredu ible. The following lemma is left as an exer ise.

Lemma A.3.5. Let A be a ring and X = Spe (A). A losed subset Y X is irredu ible if and only if I (Y ) A is a prime ideal.

We shall now de ne morphisms of spe tra. Let ' : A ! B be a ring map. Sin e for a prime ideal P ' 1 (P ) is a prime ideal in A, ' indu es a map

B , the preimage

! Spe (A) ; ! ' 1 (P ) : 1 (M ) of a maximal ideal M B need not be a

'# = Spe (') : Spe (B ) P7

Note that the preimage ' maximal ideal in A. Hen e, in general, the map '# does not indu e a map Max(B ) ! Max(A). However, if ' is integral, then '# : Max(B ) ! Max(A) is de ned by Lemma 3.1.9 (4).

Lemma A.3.6. Let ' : A ! B be a ring map. Then the indu ed map '# : Spe (B ) ! Spe (A) is ontinuous. More pre isely, we have '#

1

V (I ) = V '(I ) B .

426


Proof. Let I A be an ideal, then '#

1

V (I ) = '# 1 fP 2 Spe (A) j P I g = fQ 2 Spe B j ' 1 (Q) I g = fQ 2 Spe B j Q '(I )g = V '(I ) B :

In parti ular, the preimages of losed sets in Spe (A) are losed. Similarly to the lassi al ase ( f. Lemma A.2.13), we have the following

Lemma A.3.7. Let ' : A ! B be a ring map. Then

I '# (Spe B ) = I '# (Spe B ) =

p

Ker(') :

Proof. The rst equality follows from Lemma A.3.3, the se ond from '# (Spe B )

=

sin e, obviously, '

1

I

\

P 2Spe B p

h0i

'

=

1 (P )

='

1

\

P 2Spe B

!

P ='

1

p

h0i

;

p

Ker(').

Proposition A.3.8. Let ' : A ! B be a ring map and onsider the indu ed map '# : X = Spe B ! Spe (A) = Y .

(1) If ' is surje tive then '# (X ) = V (Ker ') and '# : X ! '# (X ) is a homeomorphism. # (X ) is dense in Y . (2) If ' is inje tive then 'p (3) Let ' : A ! Ared = A= h0i be the anoni al proje tion, then the indu ed map '# : Spe (Ared ) ! Spe (A) is a homeomorphism. Proof. (1) ' indu es an isomorphism A= Ker(') ! B , hen e, we have a bije tion Spe B ! Spe A= Ker(') . However, we also have a bije tion between prime ideals of A= Ker(') and prime ideals of A whi h ontain Ker('). This shows that '# is a bije tion X ! V Ker(') . It is also easy to see that '# and '# 1 are ontinuous. The remaining statements (2), (3) follow from Lemma A.3.7 and (1).

Note that the onverse of (1) and (2) need not be true if the rings are not redu ed: for instan e, ' : K [x℄ ,! K [x; y ℄=hy 2 i is not surje tive, but '# is a homeomorphism; ' : K [x; y ℄=hy 2 i K [x℄ = K [x; y ℄=hy i is not inje tive, but again '# is a homeomorphism. However, if '# (X ) is dense in Y , then, at least, Ker(') onsists of nilpotent elements by Lemma A.3.7. We have seen that Spe (Ared ) and Spe (A) are homeomorphi . Hen e, the topologi al spa e Spe (A) ontains less information than A | the nilpotent elements of A are invisible in Spe (A). However, in many situations nilpotent elements o

ur naturally, and they are needed to understand the situation. The notion of a s heme takes are of this fa t.


427

De nition A.3.9. The pair Spe (A); A with A a ring, is alled an aÆne

s heme . A morphism f : Spe (A); A ! Spe (B ); B of aÆne s hemes is a pair f = ('# ; ') with ' : B ! A a ring map and '# : Spe (A) ! Spe (B ) the indu ed map. The map ' is sometimes denoted as f . An isomorphism is a morphism whi h has a two{sided inverse. A subs heme of Spe (A); A is a pair Spe (A=I ); A=I , where I any ideal.

A is

Although f is determined by ' we also mention '# in order to keep the geometri language. Usually we write X = Spe (A) to denote the s heme (Spe (A); A) and sometimes we write jX j to denote the topologi al spa e Spe (A). As topologi al spa es, we have a anoni al identi ation, Spe (A=I ) = V (I ) Spe (A) ; and, usually, we shall not distinguish between V (I ) and Spe (A=I ). If A = K [x1 ; : : : ;xn ℄=I then we all A also the oordinate ring of the aÆne s heme Spe (A); A . If Xi = Spe (A=Ii ); A=Ii , i = 1; 2, are two subs hemes of the aÆne s heme X = Spe (A); A then we de ne the interse tion and the union as

X1 \ X2 := Spe (A=(I1 + I2 )); A=(I1 + I2 ) ; X1 [ X2 := Spe (A=(I1 \ I2 )); A=(I1 \ I2 ) : As topologi al spa es, we have, indeed, Spe (A=I1 + I2 ) = V (I1 ) \ V (I2 ) ;

Spe (A=I1 \ I2 ) = V (I1 ) [ V (I2 ) :

As an example, onsider I := hy 2 ; xy i K [x; y ℄. Then V (I ) A 2 onsists of the x{axis, but the aÆne s heme X = Spe (K [x; y ℄=I ); K [x; y ℄=I has an embedded fat point at 0. Sin e y 6= 0 in K [x; y℄=I , but y 2 = 0, this additional stru ture may be visualized as an in nitesimal dire tion, pointing in the y { dire tion (Fig. A.12).

6

Fig. A.12.

A line with an in nitesimal dire tion pointing out of the line.

As in the lassi al ase, we want to de ne produ ts and the graph of a morphism: let ' : C ! A and : C ! B be two ring maps, that is, A and B are C {algebras. Let X = Spe (A), Y = Spe (B ) and S = Spe (C ). Then the aÆne s heme

X S Y := Spe (A C B ); A C B ;

428


together with the proje tion maps pr1 and pr2 de ned below, is alled the bre produ t of X and Y over S . If C = K is a eld, then S = Spe (K ) is a point, and we simply set

X Y := X K Y := X Spe (K ) Y : The ring maps A ! A C B , a 7! a 1, and B ! A C B , b 7! 1 b indu e proje tion maps pr1 : X S Y

!X;

pr2 : X S Y

!Y

su h that the following diagram ommutes pr2

X S Y

/

Y

pr1

X

'#

/

#

S:

Moreover, given an aÆne s heme Z = Spe (D) and the following ommutative diagram with solid arrows, there exists a unique dotted arrow su h that everything ommutes (all arrows being morphisms of aÆne s hemes),

Z X S Y (

pr1

"

X

'#

Y /(

pr2

/

#

S:

This is alled the universal property of the bre produ t , and it is a onsequen e of the universal property of tensor produ ts (Proposition 2.7.11). Note that for A = K [x1 ; : : : ; xn ℄=I , B = K [y1 ; : : : ; ym ℄=J , X = Spe (A), Y = Spe (B ), we obtain

X Y = Spe (K [x1 ; : : : ; xn ; y1; : : : ; yn ℄=hI; J i) : Now let ' : B ! A be a C {algebra morphism. Then we have morphisms f = ('# ; ') : X ! Y and idX : X ! X and, by the universal property of the bre produ t, a unique morphism X ! X S Y su h that the diagram f X QQQQ QQQ ( X S Y idX

pr1

#

X

/( Y

pr2

/

S:


429

ommutes. Note that the morphism X ! X S Y is given on the ring level by a ring map f : A C B ! A, a b 7! a'(b). This map is surje tive and, hen e, there exists a unique aÆne subs heme f = Spe (A

C B )= Ker( f ) X S Y ;

alled the graph of f . Of ourse, sin e (A C B )= Ker f = A, the morphism X ! f is an isomorphism with 1 = pr1 j f as inverse. There is a ommutative diagram 1 X oGG f X S Y GG= yy GG 2 yyyy G pr y 2 f GG # |yy Y

with 1 and 2 the restri tions of pr1 and pr2 . Dually, we have a ommutative diagram of ring maps

f

| a7![a 1℄ / (A C B )= Ker f o o A iSSSSS = O SSS SSS ' SSSSSS SS

B:

A C mm6 m m mmm mmmb7!1 b m m m

B

If ' : B = K [y1 ; : : : ; ym ℄=IY ! A = K [x1 ; : : : ; xn ℄=IX is a morphism of aÆne K {algebras, indu ed by 'e : K [y℄ ! K [x℄, yi 7! fei , with 'e(IY ) IX , then

f : A K B = K [x; y℄=hIX + IY i

! K [x℄=IX = A

is given by [xi ℄ 7! [xi ℄, yi 7! [fei ℄, and f has as kernel J=hIX ; IY i with

J := hIX ; y1 fe1; : : : ; ym femiK [x;y℄ :

J , sin e modulo the ideal hy1 g g fe1 ; : : : ; fem = 'e(g) 2 IX , for all g 2 IY .) (Note that IY

fe1; : : : ; ym femi, we have

Hen e, in this ase we have, for K an arbitrary eld, f = Spe (K [x; y ℄=J ) :

Lemma A.3.10. Let f : X = Spe (A) ! Spe (B ) = Y be as above, then f (X ) = 2 ( f ) = V (J \ K [y1; : : : ; ym ℄) : Proof. The statement follows from Lemma A.3.7.

430


We shall explain, at the end of Se tion A.3, how this lemma generalizes Lemma A.2.18. It shows that, as topologi al spa e,

f (X ) = Spe K [y℄=(J \ K [y℄)

and, hen e, we an de ne a s heme stru ture on the losure of the image by de ning the oordinate ring of f (X ) as K [y ℄=(J \ K [y ℄). The above lemma will be used to a tually ompute equations of the losure of the image of a morphism. The ideal of f (X ) in Spe B an be de ned intrinsi ally, even for maps between arbitrary rings:

Lemma A.3.11. Let f : X = Spe (A) ! Y = Spe (B ) be a morphism indu ed by the ring map ' : B ! A. Then

f (X ) = V AnnB (A) : Proof. The proof follows easily from Lemma A.3.7, sin e Ker(') = fb 2 B j '(b)A = h0ig = 0 :B A = AnnB (A) :

Remark A.3.12. If f : X ! Y is as in Lemma A.3.10, then AnnB (A) = J \ K [y ℄ ;

that is, the stru ture de ned on f (X ) by eliminating x from J is the annihilator stru ture . Sin e annihilators are, in general, not ompatible with base

hange, one has to be areful, for example, when omputing multipli ities using ideals obtained by elimination. We illustrate this fa t by an example.

Example A.3.13. Consider the set of nine points displayed in Figure A.13.

Fig. A.13.

V (y4 y2 ; xy3 xy; x3 y xy;x4 x2 ).

Proje ting these points to the x{axis, we obtain, by eliminating y , the polynomial x2 (x 1)(x + 1), des ribing the three image points. Set{theoreti ally, this is orre t, however, it is not satisfa tory if we wish to ount

A.4 Proje tive Varieties

431

multipli ities. For example, the two border points are the image of three points ea h, hen e should appear with multipli ity three, but they appear only with multipli ity one. In ase the ideal is given by two polynomials we an use, instead of elimination, resultants (not dis ussed in this book, for a de nition see, for example, the textbooks [67, 128℄), whi h do ount multipli ities orre tly. For example, onsider the polynomials f = x(x 1)(x + 1), g = y (y 1)(y + 1), de ning together the same set as above, but all nine points being redu ed. A Singular analysis gives:

SINGULAR Example A.3.14 (elimination and resultant). ring R = 0,(x,y),dp; poly f = x*(x-1)*(x+1); poly g = y*(y-1)*(y+1); poly e = eliminate(ideal(f,g),y)[1℄; fa torize(e); //-> [1℄: // 3 linear fa tors, //-> _[1℄=1 // ea h of multipli ity 1 //-> _[2℄=x+1 //-> _[3℄=x-1 //-> _[4℄=x //-> [2℄: //-> 1,1,1,1 poly r = resultant(f,g,y); fa torize(r); //-> [1℄: //-> _[1℄=1 //-> _[2℄=x-1 //-> _[3℄=x //-> _[4℄=x+1 //-> [2℄: //-> 1,3,3,3

// 3 linear fa tors, // ea h of multipli ity 3

The resultant ounts ea h image point with multipli ity 3, as ea h of this point has three preimage points, while elimination ounts the image points only with multipli ity 1.

A.4 Proje tive Varieties AÆne varieties are the most important varieties as they are the building blo ks for arbitrary varieties. Arbitrary varieties an be overed by open subsets whi h are aÆne varieties together with ertain glueing onditions.

432


In modern treatments this glueing ondition is usually oded in the notion of a sheaf, the stru ture sheaf of the variety. We are not going to introdu e arbitrary varieties, sin e this would take us too deep into te hni al geometri

onstru tions and too far away from ommutative algebra. However, there is one lass of varieties whi h is the most important lass of varieties after aÆne varieties and almost as losely related to algebra as aÆne ones. This is the lass of the proje tive varieties. What is the dieren e between aÆne and proje tive varieties? AÆne varieties, for example C n , are in a sense open; travelling as far as we want, we an imagine the horizon | but we shall never rea h in nity. On the other hand, proje tive varieties are losed (in the sense of ompa t, without boundary); indeed, we lose up C n by adding a \hyperplane at in nity" and, in this way, we domesti ate in nity. The hyperplane at in nity an then be overed by nitely many aÆne varieties. In this way, nally, we obtain a variety overed by nitely many aÆne varieties, and we feel pretty well at home, at least lo ally. However, the importan e of proje tive varieties does not result from the fa t that they an be overed by aÆne varieties, this holds for any variety. The important property of proje tive varieties is that they are losed, hen e there is no es ape to in nity. The simplest example demonstrating this are two parallel lines whi h do not meet in C 2 but do meet in the proje tive plane P2 (C ). This is what a perspe tive pi ture suggests, two parallel lines meeting at in nity. This fa t has many important onsequen es, the most important being probably Bezout's theorem (Theorem A.8.17), whi h says that two proje tive varieties X; Y PnK (where K is an algebrai ally losed eld) of omplementary dimension, that is, dim(X ) + dim(Y ) = n, and without ommon omponent, meet in exa tly deg(X ) deg(Y ) points, if we ount the interse tion points with appropriate multipli ities. The degree deg(X ) is a global invariant and an be de ned using the Hilbert polynomial (see De nitions 5.3.3 and A.8.8), while the orre t de nition of the lo al multipli ities (for arbitrary singularities), for whi h Bezout's theorem holds, is a non{trivial task ( f. [97℄ for spe ial ases and [64℄, [147℄ for arbitrary varieties). The Bezout's theorem, and many other \proje tive theorems", do not hold for aÆne varieties and, with respe t to this geometri point of view, aÆne and proje tive varieties are ompletely dierent. From an algebrai point of view, the dieren e is not that large, at least at rst glan e. AÆne varieties in A nK are the zero{set of arbitrary polynomials in n variables, while proje tive varieties in PnK are the zero{set of homogeneous polynomials in n+1 variables. For any aÆne variety in A nK we an onsider the proje tive losure in PnK , obtained by homogenizing the ideal with the help of an extra variable. However, homogenizing an ideal is not ompletely trivial, ex ept for hypersurfa es, sin e the homogenized ideal is not generated by an arbitrary homogenized set of generators. But we an ompute generators for


433

the homogenized ideal by homogenizing a Grobner basis (with respe t to a degree ordering). Thus, we are able to ompute equations for the proje tive losure of an aÆne variety. For homogeneous ideals we an ompute the degree, the Hilbert polynomial, the graded Betti numbers and many more, all being important invariants of proje tive varieties. However, one has to be areful in hoosing the orre t ordering. Spe ial are has to be taken with respe t to elimination. This will be dis ussed in Se tion A.7. Again, let K denote an algebrai ally losed eld.

De nition A.4.1. Let V be a nite dimensional ve tor spa e over K . We de ne the proje tive spa e of V , denoted by PK (V ) = P(V ), as the spa e of lines in V going through 0. More formally, two elements v; w 2 V r f0g are alled equivalent, v w , if and only if there exists a 2 K su h that v = w . The equivalen e lasses are just the lines in V through 0 and, hen e,

P(V ) := (V r f0g)= : We denote by : (V r f0g) ! P(V ) the anoni al point v 2 V r f0g to the line through 0 and v . Pn := PnK := P(K n+1)

proje tion, mapping a

is alled the proje tive n{spa e over K . An equivalen e lass of (x0 ; : : : ; xn ) 2 K n+1 r f0g, that is, a line through 0 and (x0 ; : : : ; xn ) is alled a point of Pn , whi h we denote as

p = (x0 ; : : : ; xn ) =: (x0 : : : : : xn ) 2 Pn ; and (x0 : : : : : xn ) are alled homogeneous oordinates of p. Note that for 2 K , (x0 : : : : : xn ) are also homogeneous oordinates of p, that is, (x0 : : : : : xn ) = (x0 : : : : : xn ). Any linear isomorphism of K {ve tor spa es V = W , indu es a bije tion P(V ) = P(W ), and the latter is alled a proje tive isomorphism . In parti ular, if dimK (V ) = n + 1, then P(V ) = Pn and, therefore, it is suÆ ient to onsider n P. Note that f 2 K [x0 ; : : : ; xn ℄ de nes a polynomial fun tion f : A n+1 ! K , but not a fun tion Pn ! K sin e f (x0 ; : : : ; xn ) 6= f (x0 ; : : : ; xn ), in general. However, if f is a homogeneous polynomial of degree d, then for 2 K ,

f (x0 ; : : : ; xn ) = d f (x0 ; : : : ; xn ) sin e (x) = (x0 )0 : : : (xn )n = d x for any = (0 ; : : : ; n ) with jj = d. In parti ular, f (x0 ; : : : ; xn ) = 0 if and only if f (x0 ; : : : ; xn ) = 0

434


for all . Hen e, the zero{set in Pn of a homogeneous polynomial is well{ de ned. In the following, x denotes (x0 ; : : : ; xn ), and K [x℄d denotes the ve tor spa e of homogeneous polynomials of degree d. Note that any f 2 K [x℄ r f0g has a unique homogeneous de omposition

f = fd1 + + fdk ; fi 2 K [x℄i r f0g :

De nition A.4.2. A set X PnK is alled a proje tive algebrai set or a ( lassi al) proje tive variety if there exists a family of homogeneous polynomials f 2 K [x0 ; : : : ; xn ℄, 2 , su h that

X = V (f )2 := fp 2 PnK j f (p) = 0 for all 2 g :

X is alled the zero{set of (f )2 in Pn. As in the aÆne ase, X depends only on the ideal I = hf j 2 iK [x℄ , whi h is nitely generated, sin e K [x0 ; : : : ; xn ℄ is Noetherian. Note that the polynomials f may have dierent degree for dierent . If X = V (f1 ; : : : ; fk ) and all fi are homogeneous of degree 1, then X Pn is a proje tive linear subspa e of Pn , that is X = Pm with m = n dimK hf1 ; : : : ; fk iK . If X = V (f ) for a single homogeneous polynomial f 2 K [x0 ; : : : ; xn ℄ of degree d > 0, then X is alled a proje tive hypersurfa e in Pn of degree d. The polynomial ring K [x℄, x = (x0 ; : : : ; xn ), has a anoni al grading, where the homogeneous omponent of degree d, K [x℄d , onsists of the homogeneous polynomials of degree d (see Se tion 2.2). Re all that an ideal I K [x℄ is alled homogeneous if it an be generated

by homogeneous elements. Hen e, proje tive varieties in Pn are the zero{sets V (I ) of homogeneous ideals I K [x0 ; : : : ; xn ℄. As in the aÆne ase ( f. Lemma A.2.4) we have, using Exer ise 2.2.4,

Lemma A.4.3.

(1) ;, Pn are proje tive varieties. (2) The union of nitely many proje tive varieties is proje tive. (3) The interse tion of arbitrary many proje tive varieties is proje tive. The Zariski topology on Pn is de ned by taking as losed sets the proje tive varieties in Pn . The Zariski topology on a proje tive variety X Pn is the indu ed topology. An open subset of a proje tive variety is alled a quasi{ proje tive variety . Again, we all a proje tive variety X Pn irredu ible if it is irredu ible as a topologi al spa e, that is, if X 6= ;, and if it is not the union of two proper proje tive algebrai subsets. As in the aÆne ase, there is a tight onne tion between (irredu ible) proje tive varieties and homogeneous (prime) ideals, the reason being the proje tive Nullstellensatz.


435

De nition A.4.4. For any non{empty set X PnK de ne I (X ) := hf

2 K [x0 ; : : : ; xn ℄ j f homogeneous, f jX = 0i K [x0 ; : : : ; xn ℄ ; the (full) homogeneous (vanishing) ideal of X PnK . The quotient ring K [X ℄ := K [x0 ; : : : ; xn ℄=I (X ) is alled the homogeneous oordinate ring of X . We set

Æ

K [X ℄d := K [x0 ; : : : ; xn ℄d I (X ) \ K [x0 ; : : : ; xn ℄d : If X = V (I ) Pn is proje tive, I K [x0 ; : : : ; xn ℄ a homogeneous ideal, then the proje tive Nullstellensatz (see below) ompares I and the full homogeneous vanishing ideal I (X ) of X . Of ourse, we an re over I from X only up to radi al, but there is another ambiguity in the homogeneous ase, namely V (h1i) = ; = V (hx0 ; : : : ; xn i). We de ne

I (;) := hx0 ; : : : ; xn i =

M

d>0

K [x℄d =: K [x℄+

whi h is alled the irrelevant ideal .

Remark A.4.5. The irrelevant ideal an be used to adjust the degrees of the de ning equations of a proje tive variety. Sin e V (f ) = V (x0 f; : : : ; xn f ) Pn , we see that a proje tive variety de ned by a polynomial of degree d is also the zero{set of polynomials of degree d + 1. Hen e, if X Pn is the zero{set of homogeneous polynomials of degree di d, then X is also the zero{set of homogeneous polynomials having all the same degree d (but with more equations).

If X Pn is a proje tive algebrai set and : K n+1 r f0g ! Pn the proje tion, we may also onsider the aÆne variety

CX := 1 (X ) [ f0g = f(x0 ; : : : ; xn ) 2 A n+1 j (x0 : : : : : xn ) 2 X g [ f0g ; whi h is the union of lines through 0 in A n+1 , orresponding to the points in X if X 6= ;, and C ; = f0g. CX is alled the aÆne one of X (see the symboli pi ture in Figure A.14, respe tively Figure A.15). It is the aÆne variety in A n+1 de ned by the homogeneous ideal I K [x0 ; : : : ; xn ℄. Now onsider I (CX ), the vanishing ideal of the aÆne variety CX A n+1 . If f = fd1 + + fdk 2 I (CX ) is the homogeneous de omposition of f , then, for any (x0 ; : : : ; xn ) 2 CX r f0g, we have

f (x0 ; : : : ; xn ) = d1 fd1 (x0 ; : : : ; xn ) + + dk fdk (x0 ; : : : ; xn ) = 0 ; for all 2 K . Hen e, sin e K is algebrai ally losed and has in nitely many elements, all homogeneous omponents fd1 ; : : : ; fdk of f are in I (CX ) (this

436


CV A n+1 Fig. A.14.

The aÆne one of a proje tive variety V .

need not be true if K is nite). By Lemma 2.2.7, I (CX ) is a homogeneous ideal. Slightly more general, let Pn Pn+1 be a proje tive hyperplane, let X Pn be a proje tive variety, and let p 2 Pn+1 r Pn. Then the proje tive

one over X with vertex p is the union of all proje tive lines pq, q running through all points of X . We denote it by Cp X , it is a proje tive variety in Pn+1 . If X is given by homogeneous polynomials fi (x1 ; : : : ; xn+1 ) and if p = (1 : 0 : : : : : 0), then Cp X is given by the same polynomials, onsidered as elements of K [x0 ; : : : ; xn+1 ℄. An aÆne or a proje tive one over some variety X with vertex some point p 62 X is simply alled a one .

Theorem A.4.6 (Proje tive Nullstellensatz). Let K be an algebrai ally

losed eld and I K [x0 ; : : : ; xn ℄ a homogeneous ideal. Then

p

(1) V (I ) = ; if and only if hx0 ; :p: : ; xn i I; (2) if V (I ) 6= ; then I V (I ) = I.

Moreover, there is an in lusion reversing bije tion

proje tive algebrai sets in PnK S

irredu ible proje tive algebrai sets in PnK

S

points in P

n K

!

! !

homogeneous radi al ideals I hx0 ; : : : ; xn i

S

homogeneous prime ideals I ( hx0 ; : : : ; xn i S

homogeneous maximal ideals I ( hx0 ; : : : ; xn i

.

Proof. Let Va (I ) A n+1 , respe tively V (I ) Pn, denote the aÆne, respe tively proje tive, variety de ned by I . (1) We have V (I ) = whi h is equivalent to ; if and only if Va (I ) f0g, p hx0 ; : : : ; xn i I Va (I ) . But the latter ideal equals I , due to the usual

Nullstellensatz. (2) The onsiderations before Theorem A.4.6 imply that I Va (I ) is generated by all homogeneous polynomials vanishing on Va (I ), hen e, we have


437

I Va (I ) = I V (I ) and, therefore, everything follows from the aÆne Nullstellensatz. For example, the point (p0 : : : : : pn ) with p0 6= 0 orresponds to the ideal hp0 x1 p1 x0 ; : : : ; p0 xn pn x0 i. As an example, look at the aÆne one of the uspidal ubi , given by f = y3 + x2 z = 0. By Remark A.4.5, this is also given as the zero{set of the ideal I = hxf; yf; zf i. However, the latter ideal is not radi al. We he k this and draw the surfa e using Singular:

SINGULAR Example A.4.7 (proje tive Nullstellensatz). ring R = 0,(x,y,z),dp; poly f = x2z+y3; ideal I = maxideal(1)*f; LIB"primde .lib"; radi al(I); //-> _[1℄=y3+x2z LIB"surf.lib"; plot(f);

Fig. A.15.

//<xf,yf,zf>

// f. Figure A.15

AÆne one of uspidal ubi .

Moreover, any proje tive algebrai set X = V (I ) Pn has a unique minimal de omposition into irredu ible omponents: let P1 ; : : : ; Pr denote the minimal asso iated primes of I (whi h are homogeneous ideals by Exer ise 4.1.8), then X = V (P1 ) [ [ V (Pr ) is the unique de omposition of X into irredu ible proje tive varieties, no one ontaining another. Let us now see how to over Pn by aÆne harts.

438


De nition A.4.8. In PnK , with homogeneous oordinates (x0 : : : : : xn ), let Hi := V (xi ) = f(x0 : : : : : xn ) 2 Pn j xi = 0g denote the i{th hyperplane at in nity , i = 0; : : : ; n, and let

Ui := D(xi ) := Pn r Hi = f(x0 : : : : : xn ) 2 Pn j xi 6= 0g denote the i{th ( anoni al) aÆne hart of Pn . Usually, H0 = V (x0 ) is denoted by H1 and alled the hyperplane at in nity, and U0 is alled the aÆne

hart of Pn. In an obvious way, Hi an be identi ed with Pn 1 , and we shall see below that Ui an be identi ed with A n . Of ourse, Ui and Hi depend on the hosen

oordinates.

De nition A.4.9. For a homogeneous polynomial f 2 K [x0 ; : : : ; xn ℄, let f a (x1 ; : : : ; xn ) := f (1; x1 ; : : : ; xn ) 2 K [x1 ; : : : ; xn ℄ denote the aÆnization or dehomogenization of f . For an arbitrary polynomial g 2 K [x1 ; : : : ; xn ℄ of degree d, let

gh (x0 ; x1 ; : : : ; xn ) := xd0 g

x x1 ;:::; n x0 x0

2 K [x0 ; : : : ; xn ℄

denote the homogenization of g (with respe t to x0 ). x0 is alled the homogenizing variable .

Remark A.4.10. (1) Instead of substituting x0 = 1, we an substitute xi = 1 and obtain the aÆnization with respe t to xi . Sin e any monomial x1 1 : : : xnn of g satis es jj = 1 + + n d,

xd 0

x1 x0

1

:::

xn x0

n

= xd0 jj x1 1 : : : xnn

is a monomial of degree d. That is, g h is a homogeneous polynomial of degree d. (2) (g h )a = g for any g 2 K [x1 ; : : : ; xn ℄. (3) Let f 2 K [x0 ; : : : ; xn ℄ be homogeneous and f = xp0 g (x0 ; : : : ; xn ) su h that x0 does not divide g, then (f a )h = g. Hen e, we have x0e (f a )h = f for some e, and (f a )h = f if and only if x0 - f . For example, if f = x2 z 2 + y 3 z , and if z is the homogenizing variable, then z (f a )h = f .


439

Lemma A.4.11. Let Ui := D(xi ), then fUi gi=0;:::;n is an open overing of

Pn and

'i : Ui

! A n;

(x0 : : : : : xn ) 7

!

xb x x0 ;:::; i;:::; n xi xi xi

is a homeomorphism ( b means that the respe tive omponent is deleted). The map 'j Æ 'i 1 : A n r V (xj )

! A n r V (xi ) ; 1 (x1 ; : : : ; xn ) 7 ! (x1 ; : : : ; xi ; 1; xi+1 ; : : : ; xbj ; : : : ; xn ) x j

des ribes the oordinate transformation on the interse tion Ui \ Uj , i < j. Proof. The rst statement is lear, sin e for p = (x0 : : : : : xn ) 2 Pn there exists an i with xi 6= 0, hen e p 2 Ui . It is also lear that 'i is well{de ned and bije tive, with 'i 1 (x1 ; : : : ; xn ) = (x1 : : : : : xi 1 : 1 : xi : : : : : xn ). It remains to show that 'i and 'i 1 are ontinuous, that is, the preimages of losed sets are losed. We show this for i = 0 and set ' := '0 ; U = U0 . If Y U is losed, then there exists a losed set X = V (f1 ; : : : ; fk ) Pn , fi 2 K [x0 ; : : : ; xn ℄ homogeneous, su h that Y = U \ X . Then the image '(Y ) = V (f1a ; : : : ; fka ) A n is losed. Conversely, let W = V (g1 ; : : : ; g` ) A n be losed, gi 2 K [x1 ; : : : ; xn ℄. Sin e (gjh )a = gj , ' 1 (W ) = V (g1h ; : : : ; g`h ) \ U is losed in U .

We usually identify Ui with A n and V (xi ) with Pn . More generally, if X Pn is a proje tive variety, we identify X \ Ui with 'i (X \ Ui ) A n and, in this sense, X \ Ui is an aÆne variety. We spe ify this when we de ne morphisms of quasi{proje tive varieties in Se tion A.6.

Example A.4.12. Consider the proje tive ubi urve C in P2, given by the homogeneous equation x2 z y 3 = 0. In U0 := fx 6= 0g, we have the aÆne equation (setting x = 1), z y 3 = 0, in U1 := fy 6= 0g, we have x2 z 1 = 0 and in U2 := fz 6= 0g, x2 y 3 = 0. The line H0 := fx = 0g meets C in (0 : 0 : 1) 2 U2 , H1 := fy = 0g, meets C in (1 : 0 : 0) and (0 : 0 : 1), while H2 := fz = 0g, meets C in (1 : 0 : 0). The lo al pi tures in the neighbourhoods of 0 in U0 ; U1 ; U2 are displayed in Figure A.16. We an also represent the global urve C by a symboli pi ture in all three oordinate neighbourhoods ( f. Figure A.17). We de ne the dimension of the proje tive variety X as dim(X ) = dim(CX ) where CX is the aÆne one of X .

1;

440

A. Geometri Ba kground z

y

z

(1 : 0 : 0)

y (0 : 1 : 0)

x (0 : 0 : 1)

U0 \ C Fig.

U1 \ C

U2 \ C 3 y = 0.

2 A.16. Lo al pi tures of the proje tive ubi x z (0:1:0) (1:0:0)

x

H2 : z = 0

(0:0:1)

H1 : y = 0 Fig. A.17.

H0 : x = 0

A symboli pi ture of the proje tive ubi x2 z

y3 = 0.

Lemma A.4.13. If X Pn is a proje tive variety, then dim X = maxfdim 'i (X \ Ui ) j i = 0; : : : ; ng :

If X = V (f1 ; : : : ; fk ), fi 2 K [x0 ; : : : ; xn ℄ homogeneous, then '(X \ Ui ) is the aÆne variety in A n de ned by f1 jxi =1 ; : : : ; fk jxi =1 2 K [x0 ; : : : ; xbi ; : : : ; xn ℄, where fj jxi =1 := fj (x0 ; : : : ; xi 1 ; 1; xi+1 ; : : : ; xn ). The proof is left as an exer ise. So far, we related a proje tive variety X Pn to aÆne varieties. We did this in two dierent ways, by onsidering the aÆne one CX A n+1 and the aÆne pie es X \ Ui A n , where we xed oordinates x1 ; : : : ; xn of A n and x0 ; : : : ; xn of A n+1 , respe tively Pn. In parti ular, we have, for i = 1; : : : ; n,

X = X \ Ui

[

X \ V (xi )

where X \ Ui A n is aÆne and X \ V (xi ) Pn 1 is proje tive. Sometimes we all X \ U0 the aÆne part of X and X1 := X \ V (x0 ) the part at in nity . Now we go the opposite way and asso iate to any aÆne variety X in A n a proje tive variety X in Pn , su h that its aÆne part is equal to X .


441

De nition A.4.14. Let x0 ; : : : ; xn be homogeneous oordinates on Pn, let

U0 = D(x0 ) Pn be the aÆne hart, and let '0 : U0 ! A n be as before.

(1) Let V A n be an aÆne variety. The losure of '0 1 (V ) in Pn in the Zariski topology is alled the proje tive losure of V in Pn and denoted by V . Let V1 := V \ H1 denote the part at in nity of V (respe tively V ), where H1 := V (x0 ) denotes the hyperplane at in nity . (2) If I K [x1 ; : : : ; xn ℄ is an ideal, then

I h := hf h j f

2 I i K [x0 ; : : : ; xn ℄ is alled the homogenization of I . For g 2 K [x0 ; : : : ; xn ℄ let gjx0 =0 := g(0; x1 ; : : : ; xn ) 2 K [x1 ; : : : ; xn ℄ ; and set

I1 := gjx0 =0 g 2 I h

(3)

K [x1 ; : : : ; xn ℄ : If X Pn is proje tive then X a = '0 (X \ U0 ) A n is alled the aÆne part of X . If J K [x0 ; : : : ; xn ℄ is a homogeneous ideal, the ideal J a := J jx0 =1 := hf a j f 2 J i is alled the aÆnization of J .

The following lemma shows that I h , respe tively I1 , des ribes the proje tive

losure, respe tively the part at in nity of V (I ), and that the aÆnization J a des ribes the aÆne part of V (J ). Of ourse, an analogous result holds for J jxi =1 , the dehomogenization of J with respe t to xi . Re all that I (V ) denotes the full vanishing ideal of the aÆne, respe tively proje tive, variety V .

Lemma A.4.15. (1) Let I K [x1 ; : : : ; xn ℄ be an ideal and V = V (I ) A n . Then I (V ) = I (V )h ; V (I h ) = V and I (V1 ) = (2) If J

p

I (V )1 ; V (I1 ) = V1 :

K [x0 ; : : : ; xn ℄ is a homogeneous ideal and X = V (J ) Pn. Then I (X a ) = I (X )a ; V (J a ) = X a :

Proof. (1) By de nition, V is the interse tion of all proje tive varieties ontaining '0 1 (V ) and, hen e, the interse tion of all proje tive hypersurfa es

ontaining '0 1 (V ). Therefore, I (V ) is generated by all homogeneous polynomials f satisfying f jV = 0, that is, f a 2 I (V ). Hen e, if x0 g 2 I (V ), then g 2 I (V ).

442


This implies that I (V ) is generated by homogeneous elements f whi h are not divisible by x0 . By Remark A.4.10, for su h elements we have f = (f a )h 2 I (V )h , that is, I (V ) I (V )h . Conversely, I (V )h I (V ), sin e, by ontinuity, f h jV = 0 for f h 2 I (V )h . If I is any ideal with V (I ) = V , then V (I h ) is proje tive and ontains '0 1 (V ), hen e V V (I h ). For the other in lusion, let p 2 V (I h ). Sin e V (I h ) \ U0 = V = V \ U0 , we only have to onsider points p of the form (0 : x1 : : : : : xn ) 2 V (I h )1 . Then g h (p) = 0 for all g 2 I . Sin e I (V )m I for some m by Hilbert's Nullstellensatz, 0 = (f m )h (p) = (f h (p))m and, hen e, f h (p) = 0 for all f 2 I (V ), that is, p 2V I (V )h V by the rst part. This implies also V (I1 ) = p V1 = V I (V )1 and, hen e, by Hilbert's Nullstellenp satz I1 = I (V1 ) = I (V )1 . (2) is obvious. It an easily be seen that it is not suÆ ient to homogenize an arbitrary set of generators of I in order to obtain a set of generators of I h ( f. Example A.4.17). However, if f1 ; : : : ; fk is a Grobner basis of I with respe t to a degree ordering, then I h = hf1h ; : : : ; fkh i, f. Exer ise 1.7.4, and we have

V (I ) = V (f1h ; : : : ; fkh ) ;

V (I )1 = V (f1h jx0 =0 ; : : : ; fkhjx0 =0 ) :

In ontrast, if J = hf1 ; : : : ; fk i, with fi arbitrary homogeneous generators, then J a = hf1a ; : : : ; fka i.

Corollary A.4.16. Let V A n be an aÆne variety, V Pn its proje tive

losure and H1 := Pn r A n the hyperplane at in nity. Then

(1) V is irredu ible if and only if V is irredu ible. (2) If V = V1 [ [ Vs is the de omposition of V into irredu ible omponents, then V = V 1 [ [ V s is the de omposition of V into irredu ible

omponents. (3) The map V 7! V indu es a bije tion

faÆne varieties in A g ! n

proje tive varieties in Pn with no omponent ontained in H1

:

Proof. (1) follows, sin e I is a prime ideal if and only if I h is a prime ideal

(Exer ise 4.1.9 (1)). (2) follows easily from (1) and, using (2), we may assume in (3) that V is irredu ible. For any irredu ible proje tive variety W Pn , we have W H1 if and only if x0 2 I (W ). This implies (3), sin e x0 62 I h for any ideal I K [x1 ; : : : ; xn ℄.

Example A.4.17. Consider the ideal I := hx3 + z 2; x3 + y2 i K [x; y; z ℄. Then I h 3 z 2 y2 62 J = hx3 + z 2u; x3 + y2 ui, where u is the homogenizing variable. Indeed, V (I ) is a urve in A 3 , its proje tive losure V (I h ) should meet H1 = fu = 0g only in nitely many points, while V (J ) ontains the line fu = x = 0g as omponent in H1 .

A.5 Proje tive S hemes and Varieties

We verify this, using

443

Singular:

SINGULAR Example A.4.18 (proje tive losure). ring R = 0,(x,y,z,u),dp; ideal I = x3 + z2, x3 + y2; ideal J = homog(I,u); J; //-> J[1℄=x3+z2u //-> J[2℄=x3+y2u ideal Ih = homog(std(I),u); Ih; //the homogenization of I //-> Ih[1℄=y2-z2 //-> Ih[2℄=x3+y2u LIB "primde .lib"; minAssGTZ(J,1); //-> [1℄: //-> _[1℄=u //-> _[2℄=x //-> [2℄: //-> _[1℄=y+z //-> _[2℄=x3+z2u //-> [3℄: //-> _[1℄=y-z //-> _[2℄=x3+z2u

//the minimal asso iated primes

Note that we obtain in V (J ) the extra omponent fu = x = 0g at in nity, as J is not equal to I h . V = V (I h ) meets H1 in (0 : 1 : 1 : 0).

A.5 Proje tive S hemes and Varieties Let us give a short a

ount of proje tive s hemes , the proje tive ounterpart of aÆne s hemes.

De nition L A.5.1. Let A =

neous ideal

L

d0 Ad be a graded ring and A+ the homoged>0 Ad , whi h is alled the irrelevant ideal of A. We set

Proj(A) := fP

AjP

a homogeneous prime ideal, A+ 6 P g :

The elements of Proj(A) are alled relevant prime ideals of A, and Proj(A) is

alled the proje tive, or homogeneous, spe trum of A. For X = Proj(A) and I A a homogeneous ideal, the set

V (I ) := fP

is alled the zero{set of I in X .

2 X j P Ig

444


We have V (A+ ) = ;, V (h0i) = Proj(A), and, as for Spe (A), the sets V (I ) are the losed sets of a topology, the Zariski topology on Proj(A). This is also the indu ed topology from Proj(A) Spe (A). From now on, let R be a Noetherian ring and A = R [x0 ; : : : ; xn ℄ the polynomial ring over R with Ad the homogeneous polynomials of degree d. Then Proj(A) is alled the n{dimensional proje tive spa e over R and is denoted by PnR . If I A = R [x0 ; : : : ; xn ℄ is a homogeneous ideal, then V (I ) oin ides with Proj(A=I ) under the map sending P 2 V (I ) to its residue lass modulo I . This bije tion is even a homeomorphism, and we shall onsider Proj(A=I ) as a losed subspa e of PnR . So far, we have de ned PnR only as a topologi al spa e. To de ne a s heme stru ture on PnR or, more generally, on the losed subspa e Proj(A=I ) of PnR we annot, as we did in the aÆne ase, de ne it as a pair of the topologi al spa e Proj(A=I ) together with the homogeneous oordinate ring A=I . This would distinguish between s hemes whi h we want to be the same. Namely, if m = hx0 ; : : : ; xn i A is the irrelevant ideal, then mk I and I de ne the same zero{sets in PnR . But we also want Proj(A=I ) and Proj(A=mk I ) to de ne the same proje tive s heme. For example, we do not want to distinguish between Proj(A=m) and Proj(A=m2 ), sin e both are the empty s heme. Of ourse, Spe (A=I ) and Spe (A=mk I ) are dierent, but the dieren e is lo ated in the vertex of the aÆne one, whi h is irrelevant for the proje tive s heme. What we do is, we de ne the s heme stru ture of a proje tive variety lo ally: for f 2 A homogeneous of positive degree we set V (f ) := V (hf i) and

all its omplement

D(f ) := PnR r V (f ) a prin ipal open set in PnR . As a set, D(f ) oin ides with the homogeneous prime ideals not ontaining f . Interse ting them with

h h 2 R[x1 ; : : : ; xn ℄ homogeneous of degree n deg(f ) ; (Af )0 = fn the subring of Af of homogeneous elements of degree 0, they orrespond to all prime ideals of (Af )0 . That is, we have

D(f ) = Spe (Af )0 ; whi h is an aÆne s heme. Sin e ea h V (I ) an be written as a nite interse tion V (I ) = V (f1 ) \ : : : \ V (fr ) with f1 ; : : : ; fr a system of homogeneous generators for I , it follows that the open sets D(f ) are a basis of the Zariski topology of PnR . We de ne the s heme stru ture on PnR by the olle tion of the aÆne s hemes Spe (Af )0 , where f varies over the homogeneous elements of A of positive degree. To spe ify the s heme stru ture, we an a tually take any


445

open overing by aÆne s hemes. In parti ular, it is suÆ ient to onsider the

overing of PnR by the anoni al harts

Ui := D(xi ) = Spe (R [x0 ; : : : ; xn ℄xi )0 ; i = 0; : : : ; n, where (R [x0 ; : : : ; xn ℄xi )0 = R

x0 x x x ; : : : ; i 1 ; i+1 ; : : : ; n xi xi xi xi

is isomorphi to the polynomial ring R [y1 ; : : : ; yn ℄ in n variables over R. Now let X = Proj(A=I ) PnR be a losed subset. Then X is overed by the open subsets

D(xi ) \ X = Spe ((A=I )xi )0 ;

where

((A=I )xi )0 = R [x0 ; : : : ; xi 1 ; xi+1 ; : : : ; xn ℄=(I jxi =1 ) :

Then the s heme X = Proj(A=I ) is given by the topologi al spa e jX j and the s heme stru ture , whi h is spe i ed by the olle tion of aÆne s hemes Spe ((A=I )xi )0 , i = 0; : : : ; n. Any s heme X PnR , given in this way, is

alled a proje tive subs heme of PnR . A proje tive s heme over R is a losed subs heme of some PnR and a quasi{proje tive s heme is an open subset of a proje tive s heme with the indu ed s heme stru ture. A proje tive s heme X is alled redu ed if the aÆne s hemes Spe ((A=I )xi )0 , that is, the rings ((A=I )xi )0 are redu ed for i = 0; : : : ; n. With the above de nition, Proj(A=I ) and Proj(A=mk I ) oin ide. In parti ular, in ontrast to the ase of aÆne s hemes (where, by de nition, the ideals in A orrespond bije tively to the subs hemes of Spe (A)), there is no bije tion between homogeneous ideals in m and proje tive subs hemes of PnR , even if R is an algebrai ally losed eld. The following lemma lari es the situation and gives a geometri interpretation of saturation ( onsidered in Se tion 1.8.9).

Lemma A.5.2. Let R be a Noetherian ring, and let I; J A = R[x0 ; : : : ; xn ℄ be two homogeneous ideals. Then Proj(A=I ) and Proj(A=J ) de ne the same proje tive subs heme of PnR if and only if the saturations of I and J oin ide. Hen e, there is a bije tion proje tive subs hemes saturated homogeneous ideals ! X of PnK I m := hx0 ; : : : ; xn i

R [x0 ; : : : ; xn ℄ is sat(I ) := I : m1 = ff 2 A j mr f 2 I for some r 0g ; m = hx0 ; : : : ; xn i (see Se tion 1.8.9). I is alled saturated if sat(I ) = I . We Re all that the saturation of the homogeneous ideal I

leave the proof of the lemma as an exer ise.

446


SINGULAR Example A.5.3 (proje tive subs heme, saturation). ring R=0,(x,y,z),dp; ideal M=maxideal(1); ideal I=(x2z+y3)*M^2; I; //-> I[1℄=x2y3+x4z //-> I[2℄=xy4+x3yz //-> I[3℄=xy3z+x3z2 //-> I[4℄=y5+x2y2z //-> I[5℄=y4z+x2yz2 //-> I[6℄=y3z2+x2z3 LIB"elim.lib"; sat(I,M); //-> [1℄: //-> _[1℄=y3+x2z //-> [2℄: //-> 2

We obtain that I and hy 3 + x2 z i de ne the same subs heme of P2C . If X = Proj(A=I ) is a proje tive s heme, then Spe (A=I ) is alled the aÆne

one of X . As in the lassi al ase, we de ne the dehomogenization of a homogeneous polynomial, the homogenization of an arbitrary polynomial and the proje tive losure of an aÆne s heme. We leave the details to the reader. To summarize the dis ussion so far, we see that a proje tive s heme X PnK (K an algebrai ally losed eld) an be given algebrai ally in three dierent ways, ea h way having a learly distinguished geometri meaning.

(1) We give X by a homogeneous radi al ideal I hx0 ; : : : ; xn i. This means, by the proje tive Nullstellensatz, to give X set{theoreti ally. That is, we spe ify X as a lassi al proje tive variety or, in the language of s hemes, we spe ify X with its redu ed stru ture. (2) We give X by a homogeneous saturated ideal I hx0 ; : : : ; xn i. This is, by the above lemma, equivalent to spe ifying X as a proje tive s heme s heme{theoreti ally. (3) We give X by an arbitrary homogeneous ideal I hx0 ; : : : ; xn i. This means that we do not only spe ify X with its s heme{stru ture but we do also spe ify the aÆne one over X s heme{theoreti ally. This is sometimes alled the arithmeti stru ture of X . Note that in Se tion 4.5, respe tively Se tion 1.8.9, we have shown how the radi al, respe tively the saturation, of a polynomial ideal an be omputed ee tively.


447

So far we have not yet de ned morphisms between proje tive varieties, respe tively proje tive s hemes, sin e this is more involved than in the aÆne

ase. It is not true that su h morphisms are indu ed by morphisms of the homogeneous oordinate rings, rather we have to refer to a overing by open aÆne pie es. This will be explained in detail in Se tion A.7. Nevertheless, already, we give here the relation between varieties and s hemes in a fun torial way, referring to Se tion A.7 for the de nition of morphisms between quasi{proje tive s hemes, used in (2) below.

Theorem A.5.4. (1) There is a fun tor F from the ategory of lassi al aÆne varieties to the ategory of aÆne s hemes, de ned as follows: for X an aÆne variety, set F (X ) := Spe (K [X ℄) where K [X ℄ is the oordinate ring of X and for f : X ! Y a morphism of aÆne varieties set ' := f : K [Y ℄ ! K [X ℄ and F (f ) := ('# ; ') : Spe (K [X ℄) ! Spe (K [Y ℄) the orresponding morphism of aÆne s hemes. F has the following properties: a) F is fully faithful, that is, the map

Mor(X; Y ) ! Mor Spe (K [X ℄); Spe (K [Y ℄) ; f

7! F (f ) ;

is bije tive. In parti ular, X and Y are isomorphi if and only if the s hemes Spe (K [X ℄) and Spe (K [Y ℄) are isomorphi . b) For any aÆne variety X, the topologi al spa e jX j is homeomorphi to the set of losed points of Spe (K [X ℄) whi h is dense in Spe (K [X ℄).

) The image of F is exa tly the set of redu ed aÆne s hemes of nite type over K, that is, s hemes of the form Spe (A) with A a redu ed aÆne ring over K. (2) The fun tor F extends to a fully faithful fun tor from quasi{proje tive varieties to quasi{proje tive s hemes over K. For any quasi{proje tive variety X, the topologi al spa e jX j is homeomorphi to the (dense) set of losed points of F (X ). Irredu ible varieties orrespond to irredu ible s hemes. The image under F of the set of proje tive varieties is the set of redu ed proje tive s hemes over K. The proof of (1) is an easy onsequen e of the results of Se tion A.1 and A.2. For a detailed proof we refer to [97, Propositions II.2.6 and II.4.10℄. Note that varieties in [97℄ are irredu ible, but the generalization to redu ible varieties is straightforward. Although lassi al varieties are not s hemes in a stri t sense, however, we an use the above theorem to transfer results about s hemes to results about varieties. For example, the statement of Lemma A.3.10, saying that the losure of the image f (X ) equals V (J \ K [y1 ; : : : ; ym ℄), holds for the whole s heme, in parti ular, it holds when restri ting ourselves to the subset of losed points. We on lude that the statement of Lemma A.3.10 holds for

448


lassi al varieties, too, provided we work over an algebrai ally losed eld

K . In other words, Lemma A.2.18 is an immediate onsequen e of Lemma A.3.10.

A.6 Morphisms between Varieties The de nition of morphisms between proje tive varieties is more ompli ated than for aÆne varieties. To see this, let us make a naive try by simply using homogeneous polynomials. Let X Pn be a proje tive variety, and let f0 ; : : : ; fm be homogeneous polynomials of the same degree d in K [x0 ; : : : ; xn ℄. If p = (x0 : : : : : xn ) 2 X is a point with fi (p) 6= 0 for at least one i then f0 (p) : : : : : fm (p) is a well{de ned point in Pm. Thus, f = (f0 : : : : : fm ) de nes a map X ! Pm , provided that X \ V (f0 ; : : : ; fm ) = ;. However, by Bezout's theorem, the assumption X \ V (f0 ; : : : ; fm ) = ; is very restri tive as this is only possible if dim(X ) + dim V (f0 ; : : : ; fm ) < n. In parti ular, sin e dim V (f0 ; : : : ; fm ) n m 1, the naive approa h ex ludes morphisms X ! Pm with dim X > m. The naive approa h is, in a sense, too global. The good de nition of morphisms between proje tive and, more generally, quasi{proje tive varieties uses the on ept of a regular fun tion, where regularity is a lo al ondition (not to be onfused with regularity of a lo al ring). Sin e any aÆne variety is open in its proje tive losure, it is quasi{proje tive. Thus, we shall obtain a new de nition of morphisms between aÆne varieties whi h turns out to be equivalent to the previous one, given in Se tion A.2.

De nition A.6.1. Let X PnK and Y Pm K be quasi{proje tive varieties.

(1) A fun tion f : X ! K is alled regular at a point p 2 X if there exists an open neighbourhood U X of p and homogeneous polynomials g; h 2 K [x0 ; : : : ; xn ℄ of the same degree su h that, for ea h q 2 U , we have

f (q) =

g(q) ; h(q) 6= 0 : h(q)

f is alled regular on X if it is regular at ea h point of X . O(X ) denotes the K {algebra of regular fun tions on X . (2) A morphism f : X ! Y is a ontinuous map su h that for ea h open set V Y and for ea h regular fun tion g : V ! K the omposition g Æ f : f 1 (V ) ! K

is a regular fun tion on f

1 (V ).

Note that in (1), if g and h both have degree d, then

g(q) d g(q) g(q) = = ; h(q) d h(q) h(q)

A.6 Morphisms between Varieties

449

for all 2 K , that is, the quotient g=h is well{de ned on Pn . If X A n is quasi{aÆne, we an, equivalently, de ne f : X ! K to be regular at p, if there exists an open neighbourhood U of p in A n and polynomials g; h 2 K [x1 ; : : : ; xn ℄, not ne essarily homogeneous, su h that h(q) 6= 0 and f (q) = g(q)=h(q) for all q 2 U . If we identify K with A 1K , that is, if we endow K with the Zariski topology, then a regular fun tion is a ontinuous map X ! A 1 . Therefore, the regular fun tions X ! K are just the morphisms X ! A 1 . It is also quite easy to see that a map f : X ! Y is a morphism if and only if there exist open overings fUi g of X and fVi g of Y with f (Ui ) Vi su h that f jUi : Ui ! Vi is a morphism. It is now easy to he k that the quasi{proje tive varieties and, hen e, also the proje tive varieties, together with the morphisms de ned above, are a

ategory. This means, essentially, that the omposition of two morphisms is again a morphism of quasi{proje tive varieties. A morphism f : X ! Y between quasi{proje tive varieties is an isomorphism if it has an inverse, that is, there exists a morphism g : Y ! X su h that f Æ g = idY and g Æ f = idX . f : X ! Y is alled a losed immersion or an embedding if f (X ) is a losed subvariety of Y and if f : X ! f (X ) is an isomorphism. Note that an isomorphism is a homeomorphism but the onverse is not true, as the map A 1 ! V (x2 y 3 ) A 2 , t 7! (t3 ; t2 ) shows. So far, aÆne varieties were always given as subvarieties of some A n . Having the notion of morphisms of quasi{proje tive varieties, we extend this by saying that a quasi{proje tive variety is aÆne if it is isomorphi to an aÆne subvariety of some A n . An important example is the omplement of a hypersurfa e in an aÆne variety: if X A n is aÆne, f 2 K [x1 ; : : : ; xn ℄, then X r V (f ) is aÆne. Indeed, the proje tion A n+1 ! A n , (x1 ; : : : ; xn ; t) 7! (x1 ; : : : ; xn ), indu es an isomorphism from the aÆne subvariety V (I; tf (x) 1) A n+1 onto X r V (f ). We should like to emphasize that isomorphi proje tive varieties need not have isomorphi homogeneous oordinate rings as this was the ase for aÆne varieties. The simplest example is the Veronese embedding of P1 in P2 ,

2 : P1

! P2 ;

(x0 : x1 ) 7

! (y0 : y1 : y2 ) = (x20 : x0 x1 : x21 ) ; whi h maps P1 isomorphi to the non{singular quadri 2 (P1 ) de ned by the

homogeneous polynomial y12 y0 y1 . But the oordinate rings K [x0 ; x1 ℄ and K [y0 ; y1 ; y2 ℄=hy12 y0 y1 i are not isomorphi (show this).

Example A.6.2. More generally, the d{tuple Veronese embedding is given by d : Pn

! PN ; p 7 !

M0 (p) : : : : : MN (p) ;

450


where M0 ; : : : ; MN , N = n+d d , are all monomials x0 0 xnn in x0 ; : : : ; xn of degree jj = d, in, say, lexi ographi al order. Note that for V = V (f ) Pn a hypersurfa e of degree d, the image d (V ) PN is the interse tion of d (Pn ) with a linear hyperplane in PN .

Two proje tive varieties X; Y Pn are alled proje tively equivalent if there exists a linear automorphism of K [x0 ; : : : ; xn ℄, indu ing an isomorphism of the homogeneous oordinate rings K [X ℄ and K [Y ℄. Proje tive equivalen e is an important equivalen e relation whi h implies isomorphy, but it is stronger. As we have seen, morphisms between proje tive varieties are not as easy to des ribe as between aÆne varieties. For morphisms to A n this is, however, still fairly simple, as the following lemma shows.

Lemma A.6.3. Let X be a quasi{proje tive variety, and let Y A n be quasi{aÆne. The omponent fun tions xi : A n

! K; (x1 ; : : : ; xn ) 7 ! xi ; de ne regular fun tions xi jY : Y ! K and a map f : X ! Y is a morphism if and only if the omponent fun tion fi = xi Æ f is regular for i = 1; : : : ; n. We leave the proof of the lemma to the reader. It is now also easy to see that the anoni al morphism

'i : Ui = Pn r V (xi )

! An ;

given by (x0 : : : : : xn ) 7! (x0 =xi ; : : : ; b1; : : : ; xn =xi ), is indeed an isomorphism of aÆne varieties. Lemma A.6.3 already shows that morphisms between aÆne varieties in the sense of De nition A.2.9 (that is, represented by a polynomial map) are morphisms in the sense of De nition A.6.1. To see that both de nitions do

oin ide, we still need a better understanding of regular fun tions on an aÆne variety. Let X A n be an aÆne, respe tively X Pn a proje tive, variety, then the prin ipal open sets

D(f ) = X r V (f ) = fx 2 X j f (x) 6= 0g ;

2 K [X ℄, form a basis of the Zariski topology on X (sin e K [X ℄ is Noetherian, see also Se tion A.5). We have D(fg ) = D(f ) \ D(g ), and D(f ) is empty if and only if f = 0, due to the aÆne, respe tively proje tive, Nullstellensatz. If U X is open and f : U ! K is a regular fun tion U , then S it follows that U an be overed by nitely many prin ipal open sets, U = ki=1 D(hi ), su h that f = gi =hi on D(hi ) for some gi , hi 2 K [X ℄, where gi and hi are homogeneous of the same degree if X is proje tive. In general, it is not true that f has, on all of U , a representation as g=h with g (p) 6= 0 for all p 2 U . However, for U a prin ipal open set, this is true.

f


451

Proposition A.6.4. Let X be an aÆne (respe tively proje tive) variety,

and let h 2 K [X ℄ be non{zero (respe tively homogeneous of positive degree). Then every regular fun tion f : D(h) ! K is of the form f = g=hd on D(h), for some d > 0 and g 2 K [X ℄ (respe tively with g homogeneous of degree d deg(h), if X is proje tive). S

Proof. Let D(h) = si=1 D(hi ) su h that f jD(hi) = gi =hi . Then gi hj = gj hi on D(hi ) \ D(hj ) = D(hi hj ), hen e, we have hi hj (gi hj gj hi ) = 0 on X and gi hj = gj hi on D(hi hj ). Repla ing, for all i, hi by h2i and gi by gi hi , we may assume that gi hj S = gj hi on X , hen e this equality holds in K [X ℄. Sin e D(h) = i D(hi ), we obtain V (h) = X r D(h) = V (hp 1 ; : : : ; hs ) and the aÆne, respe tively proje tive, Nullstellensatz implies h 2 hh1 ; : : : ; hs i, P that is, hd = si=1 ki hi for some d > 0 and ki 2 K [X ℄; if X is proje tive, the ki are homogeneous of appropriate degree. P P P Set g := si=1 gi ki . Then hd gj = si=1 ki hi gj = si=1 (ki gi )hj = g hj , hen e, f = gj =hj = g=hd on D(hj ) for ea h j , that is, f = g=hd on D(h). For a further understanding of morphisms between proje tive varieties, we need to onsider rational fun tions.

De nition A.6.5. Let X be a quasi{proje tive variety. A rational fun tion on X is given by a pair (U; f ) where U X is open and dense, and f : U ! K is a regular fun tion on U . Two pairs, (U; f ) and (V; g ), are equivalent if f jW = gjW for an open dense subset W U \ V . An equivalen e lass is

alled a rational fun tion on X . The set of rational fun tions on X is denoted by R(X ).

Given a representative (U; f ) of a rational fun tion on X , there exists a maximal open set Ue su h that U Ue X and a regular fun tion fe : Ue ! K su h that fejU = f . Hen e, a rational fun tion is uniquely determined by an open dense set U X and a regular fun tion f : U ! K su h that f has no extension to a regular fun tion on a larger open set in X . U is then alled the de nition set and X r U the pole set of f . Hen e, a rational fun tion is a fun tion on its de nition set, while it is not de ned on its pole set. For example, xi =x0 is a rational fun tion on Pn , i = 1; : : : ; n, with pole set V (x0 ) and de nition set D(x0 ). We de ne the addition and multipli ation of rational fun tions on X by taking representatives. Thus, R(X ) is a K {algebra.

Theorem A.6.6. (1) Let X be a quasi{proje tive variety and X = X1 [ [ Xr the de omposition of X into irredu ible omponents. Then the map R(X )

! R(X1 ) R(Xr );

is an isomorphism of K{algebras.

f

7 ! (f jX1 ; : : : ; f jXr )

452


(2) Let X be an aÆne or a proje tive variety, and let f 2 R(X ). Then there exists a non{zerodivisor g 2 K [X ℄, homogeneous of positive degree if X is proje tive, su h that D(g) is ontained and dense in the de nition set of f and h f = d on D(g) g

for some d > 0, h 2 K [X ℄, and h homogeneous of degree d deg(g) if X is proje tive. (3) If X is an irredu ible aÆne or proje tive variety, then R(X ) is a eld. a) If X is aÆne, then R(X ) = K (X ) where K (X ) = Quot(K [X ℄) is the quotient eld of K [X ℄. b) If X is proje tive, then R(X ) = K (X )0 , where K (X )0 is the sub eld of K (X ) of homogeneous elements of degree 0, that is, K (X )0 =

f f; g 2 K [X ℄ homogeneous of same degree, g 6= 0 : g

Proof. (1) is straightforward.

(2) We use the fa t that for U X , open and dense, there exists a prin ipal open set D(g ) U , whi h is dense in X . Indeed, if U = X r A, and if X1 ; : : : ; Xr are the irredu ible omponents of X , then I (A) 6 I (Xi ) for all i, and, hen e, by prime avoidan e (Lemma 1.3.12), I (A) 6 I (X1 ) [ [ I (Xr ), that is, there is some g 2 I (A) su h that g 62 I (Xi ) for all i. Su h a g is the desired non{zerodivisor (homogeneous of positive degree if X is proje tive) and f = h=g d by Proposition A.6.4. (3) Let f=g 2 K (X ), respe tively K (X )0 , then D(g ) is dense in X , sin e X is irredu ible and, hen e, we have a map to R(X ) whi h is learly inje tive. Surje tivity follows from (2).

Note that the dimension of an irredu ible aÆne algebrai set X is equal to the trans enden e degree of the eld extension K ,! K (X ) (Theorem 3.5.1). If X is a redu ible quasi{proje tive variety with irredu ible omponents X1 ; : : : ; Xr , then the above theorem implies that R(X ) is a dire t sum of elds. More pre isely,

R(X ) = Quot(K [X1 ℄) Quot(K [Xr ℄)

for X an aÆne variety, respe tively

R(X ) = Quot(K [X1 ℄)0 Quot(K [Xr ℄)0

for X a proje tive variety, and the right{hand side is isomorphi to the total quotient ring Quot(K [X ℄), respe tively Quot(K [X ℄)0 (see Example 1.4.6 and Exer ise 1.4.8). With these preparations we an identify the ring O(X ) of regular fun tions on aÆne and proje tive varieties.


453

Theorem A.6.7. (1) Let X A n be an aÆne variety, then O(X ) is isomorphi to the aÆne

oordinate ring K [X ℄ = K [x1 ; : : : ; xn ℄=I (X ). (2) Let X Pn be a onne ted (for example, irredu ible) proje tive variety, then O(X ) = K. The theorem shows a prin ipal dieren e between aÆne and proje tive varieties: aÆne varieties have plenty of regular fun tions, while on a proje tive variety we have only regular fun tions whi h are onstant on ea h onne ted

omponent.

Proof. To see (1), note that ea h polynomial de nes a regular fun tion on X . Hen e, we have a anoni al map K [x1 ; : : : ; xn ℄ ! O(X ) with kernel I (X ). To see that the inje tive map K [X ℄ ! O(X ) is surje tive, let f 2 O(X ) and apply Proposition A.6.4 to h = 1. (2) Let X Pn be proje tive and f 2 O(X ). By Proposition A.6.4, we

an write f on the dense prin ipal open set Ui := D(xi ) as g f jUi = dii ; gi 2 K [X ℄di : xi By Theorem A.6.6, we may onsider R(X ) and, in parti ular, O(X ), as subring of the total ring of fra tions Quot(K [X ℄). Then the above equality means that f = gi =xdi i 2 Quot(K [X ℄), that is, xdi i f 2 K [X ℄di , for i = 0; : : : ; n. Ea h monomial x0 1 : : : xnn of degree N = jj d0 + + dk sati es i di for at least one i. Therefore, K [X ℄N f K [X ℄N . Multiplying, su

essively, both sides with f , we see that

K [X ℄N f q K [X ℄N for all q > 0 : Let h 2 K [X ℄ be a non{zerodivisor of degree 1 (for example, a generi linear

ombination h := 0 x0 + + n xn ), then hN f q 2 K [X ℄N for all q 0 and, hen e, 1 (K [X ℄)[f ℄ N K [X ℄ : h Sin e 1=hN K [X ℄ is a nitely generated K [X ℄{module, and sin e K [X ℄ is Noetherian, the subring (K [X ℄)[f ℄ is nite over K [X ℄ and, therfore, f satis es an integral relation (by Proposition 3.1.2)

f m + a1 f m

1+

+ am = 0;

ai 2 K [X ℄ :

Sin e f 2 Quot(K [X ℄)0 , we an repla e in this equation the ai by their homogeneous omponents of degree 0 and see that f is integral, hen e algebrai , over K [X ℄0 = K .

454


If X is irredu ible, then Quot(K [X ℄) is a eld and, sin e K is algebrai ally

losed, it follows that f belongs to K . Otherwise, the same reasoning gives that f is onstant on ea h irredu ible omponent, and the result follows from

ontinuity of f .

Morphisms in on rete terms

We an now des ribe morphisms f : X ! Y between varieties in on rete terms. If Y A m , respe tively Y Pm , then f may be onsidered as a morphism to A m , respe tively Pm , su h that f (X ) Y . We, therefore, have to distinguish two main ases, Y = A m and Y = Pm . (1) Morphisms to A m : By Lemma A.6.3, a morphism f : X ! A m is given by regular omponent fun tions fi : X ! K , i = 1; : : : ; m. If X A n is aÆne, then, by Theorem A.6.7 (1), fi = fei jX , where fei is a polynomial in K [x1 ; : : : ; xn ℄ representing fi 2 K [X ℄. If X Pn is proje tive, then f is onstant on ea h onne ted omponent of X by Theorem A.6.7 (2), and f (X ) onsists of nitely many points.

(2) Morphisms to Pm: Now onsider a map f : X ! Pm, and let Vi := D(yi ) be the anoni al harts of Pm . Then f is a morphism if and only if the restri tion f : f 1 (Vi ) ! Vi = A m is a morphism for all i = 0; : : : ; m. Hen e, we ould give a morphism f by giving morphisms gi : Ui ! Vi = A m , where fUi g is an open overing of X , su h that gi and gj oin ide on Ui \ Uj , taking into a

ount the oordinate transformation Vi = Vj . Sin e this is not very pra ti al, we prefer to des ribe morphisms to Pm by spe ifying homogeneous polynomials. To see that this is possible, assume that, for ea h irredu ible omponent Xi of X , f (Xi ) 6 H1 = V (y0 ) Pm (this an always be a hieved by a linear hange of the homogeneous oordinates y0 ; : : : ; ym of Pm ). Then f 1 (V0 ) \ Xi 6= ; and, hen e, X 0 := f 1 (V0 ) is dense in X and the restri 0 ) : X 0 ! A m with f 0 regular tion of f to X 0 is a morphism f 0 = (f10 ; : : : ; fm i 0 0 fun tions on X . Ea h fi de nes a rational fun tion fi on X and, hen e, on the proje tive losure X Pn . By Theorem A.6.6, fi is of the form hi =gidi with gi ; hi 2 K [x0 ; : : : ; xn ℄ homogeneous, deg(hi ) = di deg(gi ) and D(gi ) \ X dm and gb := g=g di . Consider the homogeneous dense in X . Set g := g1d1 : : : gm i i e e polynomials f0 := g , f1 := h1 gb1 ; : : : ; fem = hm g m , are all homogeneous of the same degree. Hen e the polynomial map fe,

x = (x0 : : : : : xn ) 7

! fe0 (x) : : : : : fem (x) ; is a morphism to Pm , de ned on Pn r V (fe0 ; : : : ; fem ), whi h oin ides on D(fe0 ) with f . Sin e D(fe0 ) = D(g1 ) \ \ D(gm ) is dense in X , f oin ides with fe on X r V (fe0 ; : : : ; fem ), the de nition set of fejX , whi h is, again, dense

in X . Sin e f is ontinuous, f is uniquely determined by fe. Hen e, any morphism X ! Pm an be given by homogeneous polynomials fe0 ; : : : ; fem of the


455

same degree, satisfying that fe0 : : : : : fem extends from X r V (fe0 ; : : : ; fem ) to a morphism de ned on X . It may, indeed, happen that X \ V (fe0 ; : : : ; fem ) 6= ; ( f. Example A.6.9). Conversely, if fe0 ; : : : ; fem are homogeneous of same degree, then, ertainly, e f = (fe0 : : : : : fem ) de nes a morphism X ! Pm if X \ V (fe0 ; : : : ; fem) = ;.

SINGULAR Example A.6.8 (morphisms of proje tive varieties).

We onsider X = V (z 3 xy 2 + y 3 ) P2 , and want to de ne a morphism : X ! P2, given by the homogeneous polynomials fe0 = xz, fe1 = xy and fe2 = x2 + yz. We have to he k that X \ V (fe0 ; fe1 ; fe2 ) = ;. ring R=0,(x,y,z),dp; ideal I=z3-xy2+y3; ideal J=xz,xy,x2+yz; dim(std(I+J)); //-> 0

From the result we read that, indeed, X \ V (fe0 ; fe1 ; fe2 ) = ;. map phi=R,J; phi(I); //-> _[1℄=x6+x3y3+3x4yz-x3y2z+3x2y2z2+y3z3

The latter means that the image of X is a urve of degree 6, given by the polynomial x6 + x3 y 3 + 3x4 yz x3 y 2 z + 3x2 y 2 z 2 + y 3 z 3 . However, if X \ V (fe0 ; : : : ; fem ) 6= ;, then fe de nes, in general, only a rational map ( f. De nition A.7.11). It de nes a morphism X ! Pm if and only if X

an be overed by open sets Uj su h that Uj fe 1 (Vj ), and su h that for ea h i, the fun tion fei =fej , whi h is regular on fe 1 (Vj ), extends to a regular fun tion on Uj (sin e fe is not de ned everywhere, the preimages fe 1 (Vj ) need not over X ). To see how this an happen, onsider (x0 : : : : : xn ) 7! (x1 : : : : : xn ), whi h de nes a morphism : Pn r fp0 g ! Pn 1 , p0 := (1 : 0 : : : : : 0). Geometri ally, this is the proje tion from a point , here p0 , to Pn 1 . A point p 2 Pn r fp0 g is mapped to the interse tion point of the proje tive line p0 p through p0 and p with the hyperplane at in nity, H1 = Pn 1 . Certainly, if n 2 then annot be extended to a ontinuous map Pn ! Pn 1 , sin e otherwise (p0 ) = (p0 p) = (p) for ea h p 2 Pn r fp0 g. If X Pn is a subvariety and p0 62 X , then the restri tion of is a morphism from X to Pn 1 . But even for p0 2 X , may de ne a morphism on all of X . We show this by a on rete example.

Example A.6.9. Let X P2 be the urve de ned by x2 y2 yz = 0, and

onsider the map de ned by X 3 (x : y : z ) 7! (x : y ) 2 P1 .

456


is the proje tion from q = (0 : 0 : 1) to the proje tive line P1 = fz = 0g. Sin e q 2 X , is not de ned at q , and we have to analyse the situation in the harts V0 = fx 6= 0g and V1 = fy 6= 0g of the image P1 . We have 1 (V0 ) = X r f(0 : 0 : 1); (0 : 1 : 1)g ; 1 (V1 ) = X r f(0 : 0 : 1)g : On 1 (V0 ) the regular fun tion y=x oin ides with

y xy yx x = 2 = = x x y(y + z ) y + z whi h is a regular fun tion on X r f(0 : 1 : 1)g, in parti ular in (0 : 0 : 1). Setting (0 : 0 : 1) := (1 : 0), we see that de nes a morphism from X to P1. On X r f(0 : 0 : 1)g, the proje tion is given by (x : y : z ) 7! (x : y) and on X r f(0 : 1 : 1)g by (y + z : x). This fa t is illustrated by the pi ture in Figure A.18.

(p) (0 : 1 : 0)

z=0

(1 : 0 : 0)

p

(0 : 0 : 1)

y=0 Fig. A.18. Proje ting proje tive line fz = 0g.

x=0

X = V (x2 y2 yz) from the point (0 : 0 : 1) 2 X to the

A.7 Proje tive Morphisms and Elimination We introdu e proje tive morphisms and prove the \main theorem of elimination theory", whi h says that the image of a losed subvariety under a proje tive morphism is again losed. Then we dis uss in some detail the geometri meaning of elimination in the ontext of proje tive morphisms and, more generally, in the ontext of rational maps. In order to be able to ompute images under proje tive morphisms, we need, as in the aÆne ase, produ ts of proje tive varieties and graphs. For simpli ity, we work in the framework of varieties, although there is no essential diÆ ulty in establishing the results for algebrai s hemes over an algebrai ally

losed eld.

A.7 Proje tive Morphisms and Elimination

457

To show that the produ t of proje tive varieties is again proje tive is less straightforward than for aÆne varieties. The easiest way is to use the Segre embedding.

De nition A.7.1. The Segre embedding of Pn Pm is the map : Pn P m (x0 : : : : : xn ); (y0 : : : : : ym ) 7

! PN ; N = (n + 1)(m + 1) 1 ; ! (x0 y0 : : : : : xi yj : : : : : xn ym ) ;

with the image of a point being the pairwise produ t of the xi and yj sorted, say, lexi ographi ally. Let zij , 0 i n, 0 j m, denote the homogeneous oordinates of PN . One an use the lo al des ription of below to he k that is inje tive and the image is n;m := (Pn Pm ), the zero{set of the quadrati equations

zij zkl

zil zkj = 0 ;

hen e, a proje tive subvariety of PN with

I (n;m) = hzij zkl

zil zkj j 0 i < k n; 0 j < l mi :

De nition A.7.2. We identify Pn Pm with the image n;m and, thus,

endow Pn Pm with the stru ture of a proje tive variety, whi h is alled the produ t of Pn and Pm . A subvariety of Pn Pm is a subvariety of PN

ontained in n;m . If f is homogeneous of degree d in zij , then we an repla e zij by xi yj and obtain a bihomogeneous polynomial in xi and yj of bidegree (d; d). Here f 2 K [x0 ; : : : ; xn ; y0 ; : : : ; ym ℄ is alled bihomogeneous of bidegree (d; e) if every monomial x y appearing in f satis es jj = d and j j = e. A bihomogeneous polynomial has a well{de ned zero{set in Pn Pm , and it follows that the losed sets in the Zariski topology or, equivalently, the proje tive subvarieties of Pn Pm are the zero{sets of bihomogeneous polynomials of arbitrary bidegrees. Indeed, by Remark A.4.5, the zero{set of a bihomogeneous polynomial of bidegree (d; e) with e < d, is also the zero{set of bihomogeneous polynomials of bidegree (d; d). For example, denote by Ui = D(xi ), i = 0; : : : ; n, respe tively Vj = D(yj ), j = 0; : : : ; m, the anoni al aÆne harts in Pn, respe tively Pm. Then the xi , respe tively the yj , are bihomogeneous polynomials of degree (1; 0), respe tively (0; 1), the produ ts xi yj are bihomogeneous of degree (1; 1), and the sets Ui Vj = Pn Pm r V (xi yj ) = A n+m are open in Pn Pm . It follows that fUi Vj gi;j is an open, aÆne overing of Pn Pm . We use the Segre embedding : Pn Pm ! PN also to de ne the produ t of two arbitrary quasi{proje tive varieties X Pn , Y Pm . Namely, the image (X Y ) is a quasi{proje tive subvariety of PN and, by identifying X Y with (X Y ), we de ne on the set X Y the stru ture of a

458


quasi{proje tive variety. Moreover, if X , respe tively Y , are losed in Pn , respe tively Pm , hen e proje tive, then the image (X Y ) is losed in PN . Therefore, X Y is a proje tive variety. In aÆne oordinates (x1 ; : : : ; xn ) on U0 Pn and (y1 ; : : : ; ym ) on V0 Pn the Segre embedding is (up to permutation of oordinates) just the map A n+m ! A N , (x1 ; : : : ; xn ; y1 ; : : : ; ym ) 7

! (x1 ; : : : ; xn ; y1; : : : ; ym ; x1 y1 ; : : : ; xn ym ) ; hen e, the image is isomorphi to the graph of the map A n+m ! A nm , (x1 ; : : : ; xn ; y1 ; : : : ; ym ) 7 ! (x1 y1 ; : : : ; xi yj ; : : : ; xn ym ) between aÆne spa es. Using the universal properties of the produ t and of the graph of aÆne varieties from Se tion A.2, respe tively A.3, one an show that the produ t of two quasi{proje tive varieties X; Y has the following universal property : the proje tions X : X Y ! X and Y : X Y ! Y are morphisms of quasi{proje tive varieties, and for any quasi{proje tive variety Z and morphisms f : Z ! X and g : Z ! Y there exists a unique morphism f g : Z ! X Y su h that the following diagram ommutes:

Z

f g

f

X YH

ww ww w ww X

w {w

X

g

HH HH Y HHHH #

Y:

Example A.7.3. The Segre map : P2 P1 ! P5, (x : y : z; s : t) 7

! (xs : xt : ys : yt : zs : zt) = (z0 : : z5 ) has as image, the Segre threefold 2;1 , given in P5 by the equations z0 z3

z1 z2 = z0 z5

z 1 z4 = z 2 z5

z 3 z4 = 0 ;

whi h may be realized as

2;1 = z 2 P5 rank zz0 zz2 zz4 < 2 : 1 3 5

If X is the quadri x2 y 2 yz = 0 in P2 , then the produ t X P1 an be de ned in P2 P1 by the two bihomogeneous polynomials s2 (x2 y 2 yz ) and t2 (x2 y 2 yz ). Hen e, (X P1 ) 2;1 P5 is given by the two additional quadrati equations z02 z22 z2 z4 = z12 z32 z3 z5 = 0.


459

As in the aÆne ase, the notion of a produ t an be generalized to the bre produ t of two morphisms. Let f : X ! S and g : Y ! S be two morphisms of quasi{proje tive varieties, then there exists a quasi{proje tive variety X S Y and morphisms pr1 : X S Y ! X , pr2 : X S Y ! Y su h that the following diagram ommutes: pr2 / X Y S

Y

X

g

pr1

f

/

S:

(X S Y; pr1 ; pr2 ) is alled the bre produ t of X and Y over S . It has the same universal property as explained in Se tion A.3, but with Z a quasi{ proje tive s heme. X S Y an be realized as a losed subvariety of X Y with pr1 and pr2 indu ed by the proje tions. In parti ular, X S Y is proje tive if X and Y are proje tive. The existen e of the bre produ t an be shown as follows. Let X Pn and Y Pm be proje tive and S = Pk (the general ase where X; Y and S are quasi{proje tive follows easily from this). Then f and g are given by homogeneous polynomials fe0 ; : : : ; fek and ge0 ; : : : ; egk where the fei , respe tively the gej , have the same degree (see the dis ussion in A.6), U = X r V (fe0 ; : : : ; fek ), respe tively V = Y r V (ge0 ; : : : ; gek ), are dense in X , respe tively Y . The set

W := (p; q) 2 U V

fe0 (p) : : fek (p) = ge0(q) : : gek (q)

is well{de ned. With respe t to the open hart fz` 6= 0g of Pk , this set is given by the equations fei =fe` = gei =ge` , i = 0; : : : ; k . Hen e, W is losed in U V and given by the bihomogeneous equations fe` egi ge` fei = 0, i; ` = 0; : : : ; k . Now de ne

\ V fei gej gei fej j 0 i; j k ; whi h is a losed subvariety of X Y , and let pr1 and pr2 be indu ed by the X S Y := X Y

proje tions to the rst, respe tively the se ond fa tor. Sin e this onstru tion oin ides with bre produ t in the aÆne ase, as de ned in Se tion A.3, the universal property an be dedu ed from that ase. We leave the details as an exer ise. The onstru tion of the spa e X S Y as above makes perfe t sense for f and g rational maps, then the above onstru tion is alled a orresponden e between X and Y ( f. [144℄). As in Se tion A.3, we obtain, as a spe ial ase of the bre produ t, the existen e of the graph of a morphism. If f : X ! Y is a morphism, then the graph of f , f := f(x; y ) 2 X Y j f (x) = y g, is a losed subvariety of X Y . The proje tion pr1 : f ! X is an isomorphism.

460


Example A.7.4. Let X P2 be the quadri x2 y2 yz = 0, and : X ! P1

the proje tion to the rst two oordinates, as in Example A.6.9. The produ t X P1 is des ribed in P2 P1 by x2 y 2 yz = 0 (bidegree (2; 0)) and by the additional equation xt ys = 0, where (x : y : z; s : t) denote the

oordinates of P2 P1 .

De nition A.7.5. Let f : X ! Y be a morphism of quasi{proje tive vari-

eties. Then f is alled a proje tive morphism if there exists a ommutative diagram X HH / Pn Y HH HH pr2 f HHH $ Y

with X ,! Pn Y as losed embedding and pr2 the proje tion. If X Pn is a proje tive variety, then ea h morphism f : X ! Y , Y a quasi{proje tive variety, is proje tive: it fa tors through the losed embedding X = f ,! Pn Y and the proje tion to the se ond fa tor, where f denotes the graph of f .

Theorem A.7.6 (Main Theorem of Elimination Theory).

Let Y be any quasi{proje tive variety and : Pn Y ! Y the proje tion on the se ond fa tor. Then is a losed map, that is, the image of any losed set in Pn Y is losed in Y . Proof. Let fUigi2I be an open aÆne overing of Y . Then A Y is losed if and only if, for all i 2 I , the interse tion A \ Ui is losed in Ui . Hen e, we may assume that Y is aÆne, that is, Y is a losed subset of some A m . But then it is obviously suÆ ient to onsider the ase Y = A m . Let X Pn A m be losed. Then X = V (f1 ; : : : ; fk ), where the fi are homogeneous polynomials of degree di in the variables x = (x0 ; : : : ; xn ), but not ne essarily homogeneous in y = (y1 ; : : : ; ym ). We show that (X ) is the zero{set of ertain determinants, hen e losed: if K [x℄d K [x℄ denotes the ve tor spa e of homogeneous polynomials of degree d, then K [x℄d K K [y℄ is a free K [y℄{module of nite rank. For d 0, de ne a K [y ℄{linear map between free modules (K [x℄i := 0 for i < 0), Ad :

k M j =1

!

K [x℄d

dj

K K [y℄ ! K [x℄d K K [y℄ K [x; y℄ ;

(g1 (x); : : : ; gk (x)) 1 7 and onsider the set

!

k X j =1

gj (x)fj (x; y) ;


Sd := y 2 Y rank Ad (y) dimK K [x℄d

461

1 ;

whi h is exa tly the set of all y 2 Y su h that Ad (y ) is not surje tive. Sin e Ad (y ) is, with respe t to xed bases, a matrix with polynomial entries, Sd is a zero{set of appropriate subdeterminants of Ad and, hen e, a

losed subset of Y . T Finally, we show that (X ) = 1 d=0 Sd . We a tually prove the equivalen e of the following four statements (for y 2 Y xed): (a) (b) ( ) (d)

y 2 (X ), that is, 1 (y) \ X 6= ;, V f1 (x; y); : : : ; fk (x; y) 6= ;, Ad (yT) is not surje tive for all d, y2 1 d=0 Sd .

The equivalen e of (a) and (b), respe tively ( ) and (d), being obvious, it remains to show the equivalen e of (b) and ( ). Clearly, for any xed y and d, the map Ad (y ) is surje tive if and only if hx0 ; : : : ; xn id hf1 (x; y ); : : : ; fk (x; y )i. On the other hand, by the proje tive Nullstellensatz, the latter is satis ed for some d if and only if the variety V f1 (x; y ); : : : ; fk (x; y ) is empty. As a orollary, we obtain

Theorem A.7.7 (image of proje tive morphisms is losed).

Let f : X ! Y be a proje tive morphism of quasi{proje tive varieties. Then the image f (X ) is a losed subvariety of Y . In parti ular, if X Pn is a proje tive variety and f : X ! Y a morphism to a quasi{proje tive variety Y , then f (X ) is a losed subvariety of Y . Note that Theorem A.7.7 provides a new proof of Theorem A.6.7 (2) about regular fun tions on a proje tive variety. Sin e any regular fun tion f on X may be onsidered as a morphism to A 1 , and sin e the losed subsets of A 1 are just nitely many points, it follows that f is onstant on ea h onne ted

omponent of X .

Geometri interpretation of elimination The above theorem allows a geometri interpretation of what we really ompute when we do elimination.

Case 1: To start with, let X Pn A m be a losed subvariety. As shown in the proof of Theorem A.7.6, X is given by an ideal I K [x; y ℄, generated by polynomials f1 ; : : : ; fk 2 K [x; y ℄, homogeneous in x = (x0 ; : : : ; xn ), and arbitrary in y = (y1 ; : : : ; ym ). Consider the proje tion : Pn A m ! A m . By Theorem A.7.7, the image (X ) is losed in A m . What do we ompute when we eliminate x0 ; : : : ; xn from I ? In general, perhaps nothing. To see this, let Va (I ) A n+1 A m be the aÆne variety de ned by I (the \aÆne one with respe t to x") and let Vp (I ) denote the \partially proje tive" variety de ned by I in Pn A m . denotes in both

462


ases the proje tion to A m . Eliminating x from I omputes I \ K [y ℄. We have V (I \ K [y ℄) = (Va (I )) A m , by Lemma A.2.13. This may be stri tly larger than the image Vp (I ) = (X ), we may even have Va (I ) = A m . Namely, Vp (I ) = Vp (hxir I ) for any r 0 (see Remark A.4.5), but sin e f0g A m Va (hxir I ) we have Va (hxir I ) = A m for r > 0. Algebrai ally, this means that I \ K [y ℄ = h0i if I is of the form hxir I 0 for some ideal I 0 with r > 0 (Lemma A.2.18). Hen e, if I = hxir I 0 , then eliminating x from I gives h0i. Nevertheless, we an ompute the image (X ) by elimination.

Lemma A.7.8. With the notations from above, (X ) = V

n \ i=0

Proof. Let Ui = fxi 6= 0g = AnAm (X ) =

n [ i=0

I jxi =1

\ K [y℄

!

:

Pn A m , then

(X \ Ui ) =

n [ i=0

(X \ Ui ) A m ;

sin e (X ) is losed, by Theorem A.7.7. Hen e, by Lemmas A.2.18 and A.2.3,

(X ) =

n [ i=0

V I jxi =1 \ K [y℄ = V

n \ i=0

I jxi =1

\ K [y℄

!

:

The following lemma implies that eliminating x from I , indeed, de nes the image (X ) if I is saturated with respe t to x0 ; : : : ; xn .

Lemma A.7.9. Let I K [x; y℄, x = (x0 ; : : : ; xn ), y = (y1 ; : : : ; ym ), be an ideal, whi h is homogeneous with respe t to x, then n \ i=0

I jxi =1

!

\ K [y℄ = (I : hx0 ; : : : ; xn i1 ) \ K [y℄ :

1 1 1 We noti e by defS that I : dhx0 ; : : : d; xn i = I : hx0 ; : : : ; xn i, the1latter being i for

ominition d0 I : hx0 ; : : : ; xn i, and that we may use I : hx0 ; : : : ; x1 n putations. Proof. If f 2 I : hxi1 \ K [y℄, then, for ea h i, xdi f 2 I for some d and, hen e, f = (xdi f jxi =1 ) 2 I jxi =1Tfor all i. Conversely, if f 2 ( ni=0 I jxi =1 ), then, for ea h i, there exists some Fi 2 I su h that f = Fi jxi =1 . By Lemma 2.2.7, ea h x{homogeneous part of Fi is in I . Hen e, multiplying the parts with appropriate powers of xi , we an assume that Fi is, indeed, x{homogeneous. But then Fi = xdi i f h(i) , where


463

the supers ript h(i) denotes the x{homogenization with xi as homogenizing variable. If, additionally, f 2 K [y ℄, then it is already x{homogeneous, and we obtain xdi i f = xdi i f h(i) 2 I for all i. This implies that, for suÆ iently large d, we have hxid f 2 I , hen e f 2 I : hxi1 \ K [y ℄. Algorithms to ompute interse tions of ideals and ideal quotients were des ribed in Se tion 1.8. Sin e these omputations may be expensive, it is worth looking for situations where they an be avoided. This happens, for example, if no irredu ible omponent of X is ontained in the hyperplane fxi = 0g, then V (I jxi =1 \ K [y ℄) = (X ). In on rete examples, it may even be worthwhile to hange oordinates instead of omputing the ideal quotient.

Remark A.7.10. The statement (and proof) of Lemma A.7.9 is even simpler if the ideal is of the form I h of some ideal I K [x1 ; : : : ; xn ; y1 ; : : : ; ym ℄, where I h denotes the homogenization with respe t to the x{variables with x0 the homogenizing variable. Then I \ K [y℄ = I h \ K [y℄ : Namely, sin e I \ K [y ℄ is x{homogeneous, I \ K [y ℄ I h . The other in lusion I h \ K [y℄ I \ K [y℄ follows, sin e I = I h jx0 =1 . Moreover, sin e xk0 f 2 I h for some k if and only if f 2 I h (by de nition of I h ), we have

I h = I h : hx0 ; : : : ; xn i1 : Combining this with Lemmas A.7.9 and A.7.8, we obtain the following: let X = V (I ) A n A m , and let X be the losure of X in Pn A m . Then

X = V (I h ) = V (I \ K [y℄) = V (I h \ K [y℄) : Re all that it is, in general, not suÆ ient to homogenize the generators of I with respe t to x in order to obtain generators for I h (see also Exer ise 1.7.4).

Case 2: Let X Pn Pm be a proje tive subvariety. Then X is given by an ideal I K [x; y ℄ generated by polynomials fi , i = 1; : : : ; k , whi h are bihomogeneous in x = (x0 ; : : : ; xn ) and y = (y0 ; : : : ; ym ), and the image (X ) Pm

is losed by Theorem A.7.6. Let Va (I ) Pn A m+1 be the aÆne one with respe t to y . It is easy to see that Va (I ) A m+1 is a one and that the orresponding proje tive variety in Pm oin ides with (X ). By Lemmas A.7.8 and A.7.9, we obtain Va (I ) = Va (I : hxi1 ) \ K [y ℄ A m+1 . Now (I : hxi1 ) \ K [y ℄ is a homogeneous ideal in K [y ℄, and we obtain

(X ) = V (I : hxi1 ) \ K [y℄ :

464


The same remark as above shows that I \ K [y ℄ may be 0, hen e, in general, it does not des ribe (X ). Remark A.7.10 applies, too.

Computing the image of proje tive morphisms and rational maps. Consider a morphism f : X ! Pm with X Pn proje tive. Then the graph n n m m f X P P P is losed and f (X ) = ( f ) P . Thus to omm pute a homogeneous ideal de ning the image f (X ) P , we an apply the

method des ribed above to an ideal de ning f in Pn Pm . However, there is a simpler method whi h we are going to des ribe now ( f. Corollary A.7.13). Any morphism f : X ! Pm an be des ribed by homogeneous polynomials f0 ; : : : ; fm 2 K [x℄, all of the same degree, subje t to some onditions whi h guarantee that, at the points x 2 X \ V (f0 ; : : : ; fm ), (f0 : : : : : fm ) an be extended to a morphism in some open neighbourhood of x (see the dis ussion at the end of Se tion A.6). Slightly more general, let us onsider rational maps.

De nition A.7.11. Let X; Y be two quasi{proje tive varieties. A rational

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Y is an equivalen e lass of pairs (U; g), where U X is open map f : X and dense and where g : U ! Y is a morphism. Two pairs (U; g ) and (V; h) are equivalent if there exists an open dense subset W U \ V su h that gjW = hjW . A rational map f : X Y is alled birational if there exist open dense sets U X , V Y su h that f : U ! V is an isomorphism.

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A rational map is denoted by a dotted arrow, sin e it is not de ned everywhere. As for rational fun tions, there exists a maximal open dense set U X su h that f : U ! Y is a morphism. f is not de ned on X r U , whi h is alled the indetermina y set or pole set of the rational map; U is alled the de nition set . Note that, in general, f annot be extended in any reasonable way to all of X , even if Y = Pm. There is, in general, no way to map points from X r U to \points at in nity". A rational map X A m is, up to a hoi e of oordinates of A m , the same as an m{tuple of rational fun tions on X . A rational map X Pm an be given either by m + 1 rational fun tions gi =hi where gi and hi are homogeneous of the same degree or, multiplying gi =hi with the least ommon multiple of h0 ; : : : ; hm , by an (m + 1){tuple (f0 : : : : : fm ), where the fi are homogeneous polynomials of the same degree, su h that X r V (f0 ; : : : ; fm ) is dense in X . Sin e the ring of rational fun tions on X is the dire t sum of the elds of rational fun tions of the irredu ible omponents of X (Theorem A.6.6), we

an give a rational map, separately, on ea h irredu ible omponent (with no

ondition on the interse tion of omponents).

99 K 99 K

99 K

Case 3: Let f0; : : : ; fm 2 K [x0 ; : : : ; xn ℄ de ne a rational map f : Pn Pm , the fi being homogeneous of the same degree, let X Pn be losed, and


465

set X0 := X r V (f0 ; : : : ; fm ). Then f (X0 ) Pm is well{de ned, though not ne essarily losed. We des ribe how to ompute the losure f (X0 ). Let I K [x0 ; : : : ; xn ℄ be a homogeneous ideal with V (I ) = X , and let Xa = Va (I ) denote the aÆne one of X in A n+1 . Moreover, let fa be the aÆne morphism (f0 ; : : : ; fm ) : A n+1 ! A m+1 . Sin e the polynomials fi are homogeneous of the same degree, it follows easily that fa (Xa ) A m+1 is a one and that fa (Xa ) r f0g = f (X0 ), where : A m+1 r f0g ! Pm denotes the

anoni al proje tion. Hen e, any homogeneous ideal in K [y0 ; : : : ; ym ℄ des ribing the losure of fa (Xa ) in A m+1 des ribes f (X0 ). By Lemma A.2.18,

fa (Xa ) = Va I; f0 y0; : : : ; fm ym

\ K [y℄

:

This does also hold if X V (f0 ; : : : ; fm ), sin e then f is not de ned anywhere on X , that is, X0 = ;, and f (X ) = ;. On the other hand, X V (f0 ; : : : ; fm ) implies hf0 ; : : : ; fm ir I for some r by the Nullstellensatz. Using the isomorphism K [x℄=I = K [x; y ℄=hI; y1 f1 ; : : : ; ym fm i, we obtain an in lusion hy0 ; : : : ; ym ir hI; y1 f1; : : : ; ym fmi \ K [y℄, hen e the latter ideal de nes the empty set in Pm . Thus, we have shown:

Proposition A.7.12. Let f = (f0 : : : : : fm ) : Pn 99 K Pm be a rational map, with fi 2 K [x0 ; : : : ; xn ℄ being homogeneous polynomials of the same degree. Moreover, let I K [x0 ; : : : ; xn ℄ be a homogeneous ideal, de ning the proje tive variety X := V (I ) Pn, and let X0 := X r V (f0 ; : : : ; fm). Then f (X0 ) = V (hI; f0 y0 ; : : : ; fm ymi \ K [y0; : : : ; ym ℄) Pm :

Corollary A.7.13. Let X = V (I ) Pn be a proje tive variety, and assume that f : X ! Pm is a morphism, then

f (X ) = V (hI; f0 y0 ; : : : ; fm ymi \ K [y0; : : : ; ym℄) : Proof. f an be des ribed by homogeneous polynomials f0 ; : : : ; fm 2 K [x℄ of the same degree, su h that X0 := X r V (f1 ; : : : ; fm ) is dense in X (see Page 454). Sin e f is ontinuous, it follows that

f (X0 ) f X0 = f (X ) ; whi h is losed in Pm and ontains f (X0 ). Hen e, f (X0 ) = f (X ), and the statement follows from Proposition A.7.12. Let us return to morphisms of aÆne varieties f : X ! A m , where X A n is aÆne. As we saw in Se tion A.2, the losure of the image f (X ) is the zero{set of the ideal J = hI; y1 f1 ; : : : ; ym fm i \ K [y ℄, where y = (y1 ; : : : ; ym ) are

oordinates of A m , f1 ; : : : ; fm 2 K [x℄, x = (x1 ; : : : ; xn ), are the omponent fun tions of f , and where I K [x℄ is an ideal des ribing X .

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Remark A.7.14. In view of the previous dis ussion, we are able to explain where the points in f (X ) r f (X ) ome from: sin e f fa tors as f : X =! f A n A m ! A m ; the image f (X ) oin ides with the image of the graph f under the proje tion . Now let f be the losure of f in Pn A m . It is easy to see that h h f = V (J ), where J is the homogenization of J with respe t to x1 ; : : : ; xn and with the new homogenizing variable x0 (see Exer ise 1.7.4 on how to

ompute J h ). By Remark A.7.10, we obtain

f (X ) = V (J \ K [y℄) = V (J h \ K [y℄) =

f

:

That is, the losure of f (X ) is the image of f under the proje tive map . Hen e, in general, the image f (X ) is not losed, be ause the preimages of points in f (X ) r f (X ) es aped to in nity, that is, to f r f = V (J h jx0 =0 ) in Pn A m . Therefore,

f

r (

V (J h jx0 =0 ) : Note that we had to take the losure of f in Pn A m , but not the losure of X in Pn . Namely, if f (X ) is not a nite set, then f : X ! A m does not extend to a morphism X ! A m , not even to X ! Pm ( onsider, for instan e, the proje tion A 2 ! A 1 ). f (X ) r f (X ) =

f)

Example A.7.15. Look at the parametrization of the uspidal ubi

A 1 ! A 2 ; t 7 ! (t2 ; t3 ) : We have f (X ) = V (x3 y 2 ) A 2 , and f = V (x t2 ; y t3 ) A 1 A 2 . In order to ompute the losure f P1 A 2 , we ompute a Grobner basis of the ideal hx t2 ; y t3 i K0 [t; 0x; y0℄ with respe t to a global monomial or-

dering satisfying t x 0 y 1 > t x 0 y 1 if > 0 . We obtain, by homogenizing this Grobner basis with respe t to t and with the homogenizing variable s, 3 f = V (x

y2 ; ty

sx2 ; tx sy; t2 s2 x) P1 A 2 :

Let us do this using Singular.

SINGULAR Example A.7.16 (proje tive elimination). ring R = 0,(t,s,x,y),(dp(1),dp); ideal I = x-t2,y-t3; //ideal of the graph of f eliminate(I,t); //-> _[1℄=x3-y2

//ideal of the losure of f(X)

ideal J = std(I);

//Groebner basis w.r.t. a good ordering

A.7 Proje tive Morphisms and Elimination J; //-> //-> //-> //->

467

J[1℄=x3-y2 J[2℄=ty-x2 J[3℄=tx-y J[4℄=t2-x

In order to homogenize only with respe t to t, with s as homogenizing variable, we map to a ring R1, where x; y are onsidered as parameters, and homogenize in this new ring: ring R1 = (0,x,y),(t,s),dp; ideal Jh = homog(imap(R,J),s); setring R; //go ba k to R ideal Jh= imap(R1,Jh); Jh; //ideal of the losure of the graph of f //-> Jh[1℄=x3-y2 //-> Jh[2℄=ty-sx2 //-> Jh[3℄=tx-sy //-> Jh[4℄=t2-s2x std(subst(Jh,t,1,s,0)); //points at infinity of the losure //-> _[1℄=1 //of the graph of f

We see that the losure of the graph of f in P1 A 2 has no points at in nity, hen e, in this ase the image f (X ) is losed. In the following simple example f (X ) is not losed: let f be the proje tion of X = V (xt 1; y ) A 3 to the (x; y ){plane: ring S = 0,(t,s,x,y),(dp(1),dp); ideal I = xt-1,y; //ideal of affine hyperbola eliminate(I,t); //-> _[1℄=y

//ideal of the proje tion

By the above, the losure of the image f (X ) equals the image of the losure of X A 3 = A 1 A 2 in P1 A 2 under the proje tion (t; x; y ) 7! (x; y ). We

ompute ideal J = std(I); ring S1 = (0,x,y),(t,s),dp; //homogenize J as ideal of ideal Jh = homog(imap(S,J),s); //polynomials in t only setring S; ideal Jh = imap(S1,Jh); Jh; //-> Jh[1℄=y //-> Jh[2℄=tx-s

//go ba k to original ring

468

A. Geometri Ba kground std(subst(Jh,t,1,s,0)); //-> _[1℄=y //-> _[2℄=x

//interse tion with infinity

Hen e, the losure of X in P1 A 2 , with oordinates (t : s; x; y ), meets in nity at (1 : 0; 0; 0) whi h is not a point of X .

Remark A.7.17. So far, we explained how to ompute the image of a map by elimination. We now pose the opposite question. Let I K [x0 ; : : : ; xn ℄ be a homogeneous ideal de ning the proje tive variety X = V (I ) Pn . What do we ompute, when we eliminate, say, x0 ; : : : ; xr 1 from I ? That is, what is the zero{set of J = I \ K [xr ; : : : ; xn ℄ ? First, noti e that J is homogeneous, that is, it de nes a proje tive variety Y = V (J ) Pn r . We laim that Y is the losure of the image of X under the proje tion from the linear subspa e H = V (xr ; : : : ; xn ) = Pr 1 to Pn r . n r Here, we de ne the proje tion from H to P as H : Pn r H

! Pn

! (xr : : : : : xn ) : Geometri ally, this is the map whi h sends p 2 Pn r H to the interse tion of V (x0 ; : : : ; xr ) = Pr , the union of all lines in Pn = Pn r with the subspa e pH r;

(x0 : : : : : xn ) 7

through p and any point of H . Similarly, we an de ne the proje tion from any other (r 1){dimensional linear subspa e to a omplementary Pn r . Sin e H de nes a rational map from Pn to Pn r , the laim is a dire t

onsequen e of Proposition A.7.12, and we obtain

V (I \ K [xr ; : : : ; xn ℄) = H (X ) : If X \ H = ;, then the proje tion H is de ned everywhere on X , hen e, de nes a morphism H : X ! Pn r , and then the image H (X ) is losed. If r = 1, then H is the point p0 = (1 : 0 : : : : : 0) and p0 is the proje tion from the point p0 to Pn 1 . That is, a point q 2 Pn r fp0 g is sent to the interse tion of the proje tive line p0 q with V (x0 ) = H1 = Pn 1 . We saw in Example A.6.9 that even if p0 2 X , p0 may de ne a morphism from X to Pn 1, and then p0 (X ) is losed in Pn 1. Note that the result may be quite dierent if we proje t from dierent points, as is shown in the following example.

Example A.7.18. Consider the twisted ubi in threespa e, whi h is the losure in P3 of the image of A 2 ! A 3 , t 7! (t; t2 ; t3 ). The image in A 3 is given by eliminating t from hx1 t; x2 t2 ; x3 t3 i:


469

SINGULAR Example A.7.19 (degree of proje tion). ring R = 0,(x(0..3),t),dp; ideal I = x(1)-t,x(2)-t2,x(3)-t3; ideal J = eliminate(I,t); J; //-> J[1℄=x(2)^2-x(1)*x(3) //-> J[2℄=x(1)*x(2)-x(3) //-> J[3℄=x(1)^2-x(2)

Hen e, the image is given by x22 x1 x3 = x1 x2 x3 = x21 x2 = 0. Homogenizing gives the ideal I = hx22 x1 x3 ; x1 x2 x3 x0 ; x21 x2 x0 i of X in P3. Proje ting X from (1 : 0 : 0 : 0) to P2, that is, eliminating x0 from I , gives the quadri polynomial x22 x1 x3 for the image, whi h has one degree less than X (see De nition A.8.8). Proje ting from (0 : 1 : 0 : 0) gives, by eliminating x1 , the ubi polynomial x32 x0 x23 , hen e, the degree of the image equals the degree of X . This is due to the fa t that (1 : 0 : 0 : 0) 2 X , while (0 : 1 : 0 : 0) 62 X . ideal Jh = homog(std(J),x(0)); eliminate(Jh,x(0)); //-> _[1℄=x(2)^2-x(1)*x(3) eliminate(Jh,x(1)); //-> _[1℄=x(2)^3-x(0)*x(3)^2

To summarize the above dis ussion, we assume that we have an ideal (arbitrary, homogeneous, partially homogeneous, or bihomogeneous) and eliminate some of the variables.

What do we ompute geometri ally when we eliminate ?

(1) Let I = hg1 ; : : : ; gk i K [x1 ; : : : ; xn ℄ and X = V (I ) A n . a) By eliminating x1 ; : : : ; xr from I , we ompute an ideal I 0 des ribing the losure of the image of X under the proje tion : A n ! A n r on the last n r oordinates (that is, I 0 = I \ K [xr+1 ; : : : ; xn ℄ and V (I 0 ) = (X )). b) Let f1 ; : : : ; fm 2 K [x1 ; : : : ; xn ℄ and f = (f1 ; : : : ; fm ) : A n ! A m , then eliminating x1 ; : : : ; xn from the ideal

J = hI; y1 f1; : : : ; ym fm i K [x1 ; : : : ; xn ; y1 ; : : : ; ym ℄

omputes the losure of the image, f (X ) A m . Moreover, if J h K [x0 ; : : : ; xn ; y1 ; : : : ; ym ℄ is the homogenization of J with respe t to x1 ; : : : ; xn and with homogenizing variable x0 , 3

3 Note that J h an be omputed from J by omputing a Grobner basis for J with 0 0 respe t to an ordering satisfying x y > x y if jj > j j. 0

470


then V (J h ) is the losure in Pn A m of V (J ) A n A m . Moreover, V (J ) = X and f (X ) = V (J h ) , where denotes the proje tion n : P A m ! A m. (2) Let I = hf1 ; : : : ; fk i, fi 2 K [x0 ; : : : ; xn ; y1 ; : : : ; ym ℄ being homogeneous in x0 ; : : : ; xn and being arbitrary in y1 ; : : : ; ym. Let X = V (I ) Pn A m and : Pn A m ! A m the proje tion. a) By eliminating x0 ; : : : ; xn from I , we ompute an ideal des ribing (Va (I )) A m , where Va (I ) A n+1 A m is the aÆne variety de ned by I . Note that (Va (I )) ontains (X ), but it may be stri tly larger. b) Eliminating x0 ; : : : ; xn from I : hx0 ; : : : ; xn i1 omputes an ideal des ribing (X ) (whi h is losed).

) If J K [x1 ; : : : ; xn ; y1 ; : : : ; yn ℄ is arbitrary and if I = J h denotes the ideal in K [x0 ; : : : ; xn ; y1 ; : : : ; yn ℄, obtained by homogenizing J with respe t to x1 ; : : : ; xn , then (X ) is des ribed by eliminating x0 ; : : : ; xn from I , that is, (X ) = V (I \ K [y1; : : : ; yn ℄). (3) If I is as in (2), but also homogeneous in yi (that is, the fi are bihomogeneous in x and y ), X = V (I ) Pn Pm 1, and the proje tion Pn Pm 1 ! Pm 1. Then a), b) and ) of (2) hold. (4) Let f0 ; : : : ; fm 2 K [x0 ; : : : ; xn ℄ be homogeneous polynomials of the same degree, let X Pn be a proje tive variety given by a homogeneous ideal I K [x0 ; : : : ; xn ℄ and let f = (f0 : : : : : fm ) : Pn Pm be the rational map de ned by the fi . Eliminating x0 ; : : : ; xn from

99 K

hI; f0

y0 ; : : : ; fm ymi K [x0 ; : : : ; xn ; y0 ; : : : ; ym ℄

omputes f (X0 ), where X0 := X r V (f0 ; : : : ; fm ). If f is a morphism, then f (X0 ) = f (X ). (5) Let I K [x0 ; : : : ; xn ℄ be a homogeneous ideal and X = V (I ) Pn . ElimPn r is inating x0 ; : : : ; xr 1 from I omputes H (X ), where H : Pn r 1 the proje tion from H = V (xr ; : : : ; xn ) = P . If X \ H = ;, then H is a morphism and the image H (X ) is losed in Pn r .

99 K

A.8 Lo al versus Global Properties So far, we have dis ussed global properties of aÆne or proje tive varieties. Algebrai ally, these properties are oded in the aÆne or homogeneous oordinate ring. When talking about lo al properties, we mean those properties whi h hold in an arbitrary small neighbourhood of a given point. Sin e any proje tive variety is overed by open aÆne varieties, we an, without loss of generality, restri t our attention to aÆne varieties. However, of spe ial interest are results whi h ombine lo al and global properties.

A.8 Lo al versus Global Properties

471

Given an aÆne variety X and a point p 2 X , then the neighbourhoods of p in X are omplements of losed subvarieties not ontaining the point p. Re all from Se tion A.6 that the oordinate ring of X is isomorphi to O(X ), the ring of regular fun tions on X . Restri ting the regular fun tions to smaller and smaller neighbourhoods of p, we obtain, in the limit, a ring of germs at p of regular fun tions, the lo al ring of X at p. It turns out that this ring is just the lo alization of the oordinate ring at the maximal ideal de ning the losed point p. As we have seen in Chapter 1, we an ee tively

ompute in su h rings. If X = Spe (A) is an aÆne s heme, and m 2 X a maximal ideal of A, then the lo alization Am is the straightforward generalization of the lo al ring of a lassi al variety. Hen e, Am should be onsidered as the ring arrying information about X , whi h is valid in an arbitrary small neighbourhood of the losed point m in X , in the Zariski topology. So far we onsidered \lo al" with respe t to the Zariski topology, whi h is a rather oarse topology. When working with omplex varieties or s hemes we also have the (usual) Eu lidean topology. The lo al rings arising from this topology are dis ussed in the next se tion on singularities. In this se tion, a variety means a lassi al quasi{proje tive variety. Eventually, we mention the orresponding de nitions and statements for arbitrary quasi{proje tive s hemes. The following de nition applies to any variety.

De nition A.8.1. Let X be a variety and p a point of X .

A germ of a regular fun tion at p is an equivalen e lass of pairs (U; f ) where U is an open neighbourhood of p and f a regular fun tion on U . Two su h pairs (U; f ) and (V; g ) are equivalent if there exists an open neighbourhood W U \ V of p su h that f jW = g jW . The lo al ring of X at p is the ring of germs of regular fun tions of X at p and is denoted by OX;p . We set dimp X := dim OX;p . X is alled pure dimensional or equidimensional if dimp X is independent of p 2 X .

Proposition A.8.2. Let X A n be an aÆne variety with oordinate ring

K [X ℄ = K [x1 ; : : : ; xn ℄=I (X ), and let p 2 X, then the following holds: (1)

OX;p is a lo al ring with maximal ideal mp := mX;p := ff 2 OX;p j f (p) = 0g ;

the vanishing ideal of p. (2) OX;p = K [X ℄mp . (3) dim OX;p = maxfdim Xi j p 2 Xi g where the Xi are the irredu ible omponents of X. Proof. (1) Let f be a regular fun tion with f (p) 6= 0. Then, sin e f is ontinuous, f (q ) 6= 0 for all q in a neighbourhood U of p. Hen e, (U; 1=f ) is an

472


inverse of (U; f ) and, therefore, every element f 2 OX;p r mp is a unit and mp is the only maximal ideal of OX;p . To see (2), note that we have OX;p = f(U; f ) j f regular on U g= , and

f f; g 2 K [X ℄; g(p) 6= 0 : K [X ℄mp = g Now, let f=g 2 K [X ℄mp . Sin e g is ontinuous, g 6= 0 in a neighbourhood U of p, hen e, f=g de nes a regular fun tion on U . Therefore, we obtain a map K [X ℄mp ! OX;p , whi h is easily seen to be bije tive. (3) The anoni al map j : K [X ℄ ! K [X ℄mp , f 7! f=1, indu es a bije tion between the set of prime ideals in K [X ℄mp and the set of all prime ideals in K [X ℄ whi h are ontained in mp , via P 7! j 1 (P ) and Q 7! QK [X ℄mp . Hen e, dim K [X ℄mp equals the maximal length of hains of prime ideals in K [X ℄ ontained in mp . But ea h maximal hain of prime ideals starts with a minimal asso iated prime I0 of I (X ), whi h orresponds to an irredu ible

omponent of X . The latter ontains p, sin e I0 mp .

Example A.8.3. Consider the redu ible variety X de ned by zx = zy = 0,

onsisting of the line fx = y = 0g and the plane fz = 0g (see Figure A.19). (0; 0; 1) (0; 0; 0)

Fig. A.19.

(1; 0; 0)

The variety V (hxz;yz i).

Let 0 := (0; 0; 0), q := (1; 0; 0) and p := (0; 0; 1), then the geometri pi ture (Figure A.19) suggests dim0 X = dimq X = 2 and dimp X = 1. We he k this for the respe tive lo al rings: OX;0 = K [x; y; z ℄hx;y;zi=hxz; yz i, whi h is a lo al ring of dimension 2. As x is a unit in K [x; y; z ℄hx 1;y;zi, the lo al ring OX;q is isomorphi to K [x; y; z ℄hx 1;y;zi=hz i, whi h has dimension 2. In K [x; y; z ℄hx;y;z 1i we have z as a unit, hen e, OX;p = K [x; y; z ℄hx;y;z 1i=hx; y i has dimension 1.

SINGULAR Example A.8.4 (lo al and global dimension).

We on rm the latter by using Singular:

ring R = 0,(x,y,z),dp; //global affine ring ideal I = xz,yz;

A.8 Lo al versus Global Properties dim(std(I)); //-> 2

473

//dimension of affine variety V(I)

ring r = 0,(x,y,z),ds; //lo alization of R at (0,0,0) ideal I = imap(R,I); dim(std(I)); //-> 2 //dimension of V(I) at (0,0,0) map phi1 = r,x-1,y,z; ideal I1 = phi1(I); dim(std(I1)); //-> 2

//maps the point (1,0,0) to (0,0,0)

map phi2 = r,x,y,z-1; ideal I2 = phi2(I); dim(std(I2)); //-> 1

//maps the point (0,0,1) to (0,0,0)

//dimension of V(I) at (1,0,0)

//dimension of V(I) at (0,0,1)

For proje tive varieties, we an des ribe the lo al rings also dire tly with the help of the homogeneous oordinate ring.

Lemma A.8.5. Let X Pn be a proje tive variety, let p 2 X be a point,

and let K [X ℄ = K [x0 ; : : : ; xn ℄=I (X ) be the homogeneous oordinate ring of X. Then

OX;p = K [X ℄(mp) :=

f f; g 2 K [X ℄ homogeneous of g the same degree, g(p) 6= 0

:

We leave the proof, whi h is similar to the aÆne ase, as an exer ise. The above statements are used to de ne lo al rings for s hemes.

De nition A.8.6. Let X be a quasi{proje tive s heme, p 2 X , and let

U = Spe (A) be an aÆne neighbourhood of p. If P denotes the prime ideal of A orresponding to p, then the lo al ring of X at p is OX;p := AP , the lo alization of A at P . Of ourse, p = P , but we have hosen dierent letters to distinguish between geometry and algebra.

Let A = K [x1 ; : : : ; xn ℄=I be an aÆne K {algebra, K algebrai ally losed and X = Spe (A). Then any losed point p orresponds to a maximal ideal mp , whi h is of the form mp = hx1 p1 ; : : : ; xn pn i=I (Corollary A.2.7) and, hen e, A=mp = K . Therefore, any f 2 A determines a fun tion

f losed points of X g = Max(A) ! K ; p 7 ! f (p) := lass of f in A=mp :

474


However, due to nilpotent elements in A, p 7! f (p) may be the zero{fun tion, even if f 6= 0 in A. The simplest example is A = K [x℄=hx2 i with 0 as the only

losed point, m0 = hxi, x(0) = 0, but x 6= 0 in A. Hen e, if X is a non{redu ed s heme, we annot de ne the lo al rings as germs of ontinuous fun tions. Indeed, sin e Max(A) is dense in Spe (A), every ontinuous fun tion whi h is 0 on Max(A) must be 0 on Spe (A), identi ally. Lo al properties of varieties are of parti ular interest in the neighbourhood of singular points.

De nition A.8.7. Let X be a variety or s heme and p 2 X . Then p is alled a singular point of X , or X is alled singular at p, if the lo al ring OX;p is not a regular lo al ring. Otherwise, p is alled a regular point , or non{singular point of X . X is alled regular if it is regular at ea h point p of X . The de nition applies, if X is a s heme, to losed and non{ losed points. Re all that a lo al ring A with maximal ideal m is regular, if dim(A) = edim(A), where edim(A) = dimA=m (m=m2 ) is the embedding dimension. This is the de nition whi h works in general, unfortunately, it is not very geometri . A good geometri interpretation of regular points is given by the Ja obian riterion ( f. Corollary 5.6.14), whi h is valid for varieties over an algebrai ally losed eld. If X A n is aÆne with ideal I (X ) = hf1 ; : : : ; fk i and p 2 X , then X is regular at p if and only if dimp X = n

rank

fi (p) : xj

That is, the linear parts of the power series expansions of f1 ; : : : ; fk at p de ne a linear subspa e of the same dimension as X at p. By the impli it fun tion theorem for formal power series (Theorem 6.2.17), there exists an analyti oordinate hange at p, that is, an automorphism ' of the power series ring K [[x1 ; : : : ; xn ℄℄ (but, in general, not of the polynomial ring) su h that ' I (X )K [[x1 ; : : : ; xn ℄℄ = hx1 ; : : : ; xr iK [[x1 ; : : : ; xn ℄℄, where r is the rank of the Ja obian matrix at p. This means that X is regular at p if and only if formally at the point p; X is analyti ally isomorphi to a linear subspa e. We shall dis uss regular and singular points in the framework of lo al analyti geometry in the next se tion on singularities. The Ja obian riterion is also the basis for omputing the singular lo us

Sing(X ) = p 2 X X is singular at p

of a variety X . If X is equidimensional, this is parti ularly simple. We just fi , see have to ompute the (n dim X ){minors of the Ja obian matrix x j Corollary 5.7.5.


475

If X is not equidimensional, then Sing(X ) is the union of the singular lo i of the irredu ible omponents of X and the lo us of the pairwise interse tions of the omponents. Hen e, Sing(X ) an be omputed from a primary de omposition of X and then applying the Ja obian riterion to ea h irredu ible

omponent. In pra ti e, it is, however, heaper to ompute an equidimensional than a prime de omposition. Methods to ompute an ideal des ribing the singular lo us without any de omposition by using either tting ideals or Ext ( f. Exer ises 7.3.23 and 7.3.24) also exist. Algorithms, des ribing the singular lo us, are given in Chapter 5, Se tion 5.7. We are parti ularly interested in distinguishing singular points by invariants. Invariants an be numbers, groups, ve tor spa es or other varieties whi h re e t ertain properties of the variety at p and whi h do not hange under isomorphisms of the lo al ring OX;p . Ideally, one ould hope to lassify all singular points on a given variety by a dis rete set of invariants. However, this is only possible for the most simple singularities, the ADE{singularities (see Se tion A.9). One warning is perhaps in order. One should not expe t the lassi ation of singularities to be simpler than the lassi ation of, say, proje tive varieties. For example, if X Pn is proje tive, then the aÆne one CX A n+1 has a singularity at 0 and a lassi ation of this singularity, up to lo al isomorphism (whi h must be linear as CX is a one), implies a lassi ation of X up to proje tive equivalen e. Hen e, the lassi ation of singularities in ludes the proje tive lassi ation of proje tive varieties, but the singularities arising this way are only the homogeneous ones, whi h is a small sub lass of all singularities. In this se tion we dis uss only one lo al invariant, the multipli ity, and ompare it with a global invariant, the degree. The multipli ity was introdu ed in Se tion 5.5, using the Hilbert{Samuel fun tion. This is, perhaps, algebrai ally the most elegant way but not very geometri . Algebrai ally, as well as geometri ally, the multipli ity is the lo al

ounterpart of the degree, see Se tion 5.3.

De nition A.8.8. (1) Let X be any variety and p 2 X . The multipli ity of X at p is the multipli ity of the lo al ring OX;p in the sense of De nition 5.5.2, and denoted by mult(X; p). (2) Let X Pn be a proje tive variety, then the degree of X is the degree of the homogeneous oordinate ring K [X ℄ in the sense of De nition 5.3.3, and denoted by deg(X ) . Note that mult is invariant under lo al isomorphisms of varieties (that is, isomorphisms of their lo al rings), while deg is not invariant under isomorphisms of proje tive varieties. The degree is only invariant under proje tive equivalen e or, geometri ally speaking, it is an invariant of the embedding.

476


Example A.8.9. Consider the d{tuple Veronese embedding of Example A.6.2. One shows that d is an isomorphism from Pn onto the image Vd := d (Pn ), whi h is alled the Veronese variety . The hypersurfa e M0 (x) := fxd0 = 0g in Pn is mapped isomorphi ally onto the hyperplane fz0 = 0g in PN . M0 (x) has degree d, fz0 = 0g degree 1. Moreover, one an show that Vd has degree dn in PN , while Pn ertainly has degree 1 in Pn.

It is useful to ompare the notions of degree and multipli ity for the most simple ase, the ase of a hypersurfa e. If f 2 K [x1 ; : : : ; xn ℄ is a squarefree polynomial with f (0) = 0, and if X A n is the hypersurfa e de ned by f = 0, then mult(X; 0) = ord(f ), the smallest degree of a monomial appearing in f (Corollary 5.5.9), while deg(X ) = deg(X ) = deg(f ), the largest degree of a monomial appearing in f (Lemma 5.3.5). Here, X Pn is the proje tive

losure of X , de ned by f h = 0, where f h 2 K [x0 ; : : : ; xn ℄ denotes the homogenization of f . We start with a geometri interpretation of the degree as the number of interse tion points with a suÆ iently general hyperplane. What is meant here by \suÆ iently general" will be explained in the proof.

Proposition A.8.10. Let X Pn be a proje tive variety of dimension d,

and let X1 ; : : : ; Xr ; Xr+1; : : : ; Xs be the irredu ible omponents of X, ordered su h that dim Xi = d for i = 1; : : : ; r and dim Xi < d for i = r + 1; : : : ; s. Then deg(X ) = #(X \ H ) =

r X i=1

#(Xi \ H ) ;

where H Pn is a suÆ iently general proje tive hyperplane of dimension n d and # denotes the ardinality. P

Proof. By Lemma 5.3.11, we have deg(X ) = ri=1 deg(Xi ). Hen e, we an assume X to be irredu ible with homogeneous oordinate ring K [x0 ; : : : ; xn ℄=P , where P is a homogeneous prime ideal. Choose a homogeneous Noether normalization X ! Pd (Theorem 3.4.1): after a linear hange of oordinates, we have an in lusion

K [x0 ; : : : ; xd ℄ K [x0 ; : : : ; xn ℄=P

together with irredu ible homogeneous polynomials gd+1 ; : : : ; gn 2 P , pi 1 pi + X (x ; : : : ; x )xj ; gi = xi ij 0 i 1 i j =0

su h that Quot(K [x0 ; : : : ; xn ℄=P ) = Quot(K [x0 ; : : : ; xd ℄)[xd+1 ℄=hgd+1 i, espe ially, pd+1 = deg(X ), by Proposition 5.3.10. Set f := gd+1 , whi h de nes a hypersurfa e V (f ) Pd+1 of degree deg(X ). Then, for a suÆ iently general line L, the restri tion f jL has exa tly deg(X )


477

simple roots, that is, the interse tion V (f ) \ L onsists of pre isely deg(X ) points. We show now that the proje tion : X ! V (f ) is birational, and that the preimage of a general line, H = 1 (L) interse ts X in deg(X ) points. Let 2 K [x0 ; : : : ; xd ℄ be the dis riminant of f (with respe t to xd+1 ). is homogeneous and vanishes at a point a = (a0 : : : : : ad ) 2 Pd if and only if f (a0 ; : : : ; ad; xd+1 ) has a multiple root. 4 By Lemma 3.5.12 (applied as in the proof of Theorem 3.5.10), we have an in lusion 1 K [x0 ; : : : ; xn ℄=P K [x0 ; : : : ; xd+1 ℄=hf i :

In parti ular, there exist polynomials qd+2 ; : : : ; qn 2 K [x0 ; : : : ; xd+1 ℄ su h that xi qi 2 P , for i = d + 2; : : : ; n, and

hf; xd+2

qd+2 ; : : : ; xn qn iK [x0 ; : : : ; xn ℄ = P K [x0 ; : : : ; xn ℄ :

Hen e, for any point (x0 : : : : : xd ) 2 Pd r V () we have (x0 : : : : : xn ) 2 V (P )

()

f (x0 ; : : : ; xd+1 ) = 0 and q (x ; : : : ; xd+1 ) ; i = d + 2; : : : ; n : xi = i 0 (x0 ; : : : ; xd )

This shows that the proje tion : Pn ! Pd+1 indu es an isomorphism

: X r V ()

=

! V (f ) r V () : Therefore, if a = (a0 : : : : : ad ) 2 Pd r V (), then the line L := f(a : xd+1 ) j xd+1 2 K g interse ts V (f ) in deg(X ) distin t points, and the hyperplane

H := 1 (L) = f(a : xd+1 : : : : : xn ) j xd+1 ; : : : ; xn 2 K g interse ts X in exa tly deg(X ) points, too. In the above proof we showed, moreover,

Proposition A.8.11. Let X Pn be an irredu ible d{dimensional proje -

tive variety, then, for general homogeneous oordinates x0 : : : : : xn , there exists a hypersurfa e V (f ) Pd+1 = V (xd+2 ; : : : ; xn ) su h that the proje tion : X ! V (f ) is birational. Moreover, deg(X ) = deg(f ). 4 Another way to des ribe the zero{set of the dis riminant is given by onsidering the proje tion V (f ) ! Pd. Then V () is the image of the set of riti al points C = V (f; f=xd+1 ) in Pd (whi h we know already to be losed by Theorem A.7.7). Note that the polynomial 2 K [x0 ; : : : ; xd ℄, obtained by eliminating xd+1 from hf; f=xd+1 i, has the same zero{set as and satis es hi h i, however, the in lusion may be stri t. 0

0

478


SINGULAR Example A.8.12 (degree of proje tive variety).

Consider the rational normal urve C of degree r in Pr , whi h is the proje tive

losure of the image of the morphism

P1 3 (s : t) 7 ! (sr : sr

1t

: : : : : t r ) 2 Pr :

The homogeneous ideal of C is the kernel of the ring map

K [x0 ; : : : ; xr ℄

! K [s; t℄;

xi 7

! sr

i ti :

We ompute the degree of C and ount the number of interse tion points of C with a general hyperplane: LIB"random.lib"; int r = 5; ring R = 0,x(0..r),dp; ring S = 0,(s,t),dp; ideal I = maxideal(r); //s^r, s^(r-1)*t,..., s*t^(r-1), t^r ideal zero; map phi = R,I; //R --> S, x(i) --> s^(r-i)*tî setring R; ideal I = preimage(S,phi,zero); //kernel of map phi R --> S I = std(I); //ideal of rational normal urve C dim(I); //dimension of affine one over C //-> 2 degree(I); //degree of C is 5 //-> 5 ideal L = sparsepoly(1,1,0,10); //a general linear form //?sparsepoly; explains the syntax ideal CL = std(I+L+(x(0)-1)); //ideal of interse tion of C //with L=0 //in affine hart x(0)=1 vdim(CL); //number of interse tion points is 5 //-> 5 //in affine hart x(0)=1

Note that vdim ounts the number of interse tion points with multipli ity. By Proposition A.8.10, this number should oin ide with the degree of C . In order to ount the points set theoreti ally, we ompute the radi al of CL. Sin e C is smooth ( he k this), a general hyperplane meets C in simple points, hen e CL should oin ide with the radi al of CL and vdim should give the same number. We he k this: LIB"primde .lib"; vdim(std(radi al(CL))); //-> 5


479

Let us now onsider the multipli ity. It has a similar geometri interpretation as the degree. It ounts the number of interse tion points of a variety with a general hyperplane of the right dimension in a suÆ iently small neighbourhood of the given point. However, this interpretation is only valid in the Eu lidean topology, not in the Zariski topology. The algebrai reason for this fa t is that the Noether normalization of an algebrai lo al ring may fail. The Noether normalization holds for aÆne algebras (Theorem 3.4.1) and for analyti algebras (Theorem 6.2.16) but, in general, not for the lo alization of an aÆne algebra (Exer ise 3.4.8). Therefore, we assume in the following dis ussion that K = C , and we use the Eu lidean topology .

Proposition A.8.13. Let X C n be a variety of dimension d and p 2 X.

We denote by X1 ; : : : ; Xr ; Xr+1 ; : : : ; Xs the irredu ible omponents of X passing through p, su h that dim Xi = d for i = 1; : : : ; r and dim Xi < d for i = r + 1; : : : ; s. Then, for any suÆ iently small (Eu lidean) neighbourhood U of p and for any suÆ iently general hyperplane H C n of dimension n d and suÆ iently lose to p (but not passing through p), we have mult(X; p) = #(X \ H \ U ) =

r X i=1

#(Xi \ H \ U ) :

The proof is ompletely analogous to the previous one, by using an analyti Noether normalization ( f. Theorem 6.2.16, Exer ise 6.2.1 for the formal

ase). Moreover, we need a Weierstra polynomial for generi oordinates,

f = xpd+1 +

p 1 X j =0

j (x1 ; : : : ; xd )xjd+1 ;

where j (x1 ; : : : ; xd ) are onvergent power series and, thus, de ne holomorphi fun tions in a neighbourhood of (p1 ; : : : ; pd ) 2 C d . This follows from the Weierstra preparation theorem for onvergent power series ( f. [87℄, [104℄, for example). Now everything else works as in the global ase.

Note that mult(X; p) = mult Cp (X ); p , where Cp (X ) is the tangent one of X at p, whi h is de ned as follows: if X C n is de ned by an ideal I C [x1 ; : : : ; xn ℄, then Cp (X ) is de ned by Inp (I ). Here Inp (I ) denotes the ideal generated by the initial forms of all f 2 I , written as polynomials in y := x p ( f. Proposition 5.5.12). The tangent one an be omputed a

ording to Lemma 5.5.11. Geometri ally, the tangent one is the union of all limits of se ants ppi with pi 2 X r fpg a sequen e of points onverging to p. The dire tion of the hyperplane H in Proposition A.8.13 (equivalently, the hoi e of the generi oordinates) is predi ted by the tangent one. Choose a hyperplane H0 through p, whi h is transversal to Cp (X ), that is, H0 \ Cp (X ) = fpg. Then a small,

480


suÆ iently general displa ement of H0 interse ts U \ X in exa tly mult(X; p) points.

Example A.8.14. Let X be the uspidal ubi given by f := x3 y2 = 0, and 3 y2 has two let L" := fx + by = "g be a general line. Then f jL" = " by zeros lose to 0 ( he k this), and we have ord(f ) = mult(X; 0) = 2 ( f. Figure A.20). The tangent one is the x{axis fy = 0g, and the line L" is a small displa ement of the line L0 = fx + by = 0g, whi h is transversal to the tangent

one. A small displa ement of the line fy = 0g meets X in three omplex points ( he k this), although we see only one point in the real pi ture.

0

L0 Fig. A.20.

X L mult(X; 0) = #(L" \ X ) = 2.

SINGULAR Example A.8.15 (multipli ity and tangent one).

Although the above example is very simple, let us demonstrate the ommands to ompute the multipli ity and the tangent one. Moreover, we ompute the interse tion multipli ity of the urve ff = 0g with a general line Lo through 0 (whi h is transversal to the tangent one) and with the spe ial line L1 (whi h

oin ides with the tangent one). In order to ompute the orre t multipli ity, we have to work with a lo al degree ordering (the pro edure tangent one works for any ordering). ring r = 0,(x,y),ds; ideal f = x3-y2; mult(std(f)); //-> 2 LIB"sing.lib"; tangent one(f); //-> _[1℄=y2

//the multipli ity of f at 0

//the tangent one of f at 0

ideal Lo = random(1,100)*x + random(1,100)*y;


481

vdim(std(f+Lo)); //-> 2

//a general line through 0 //interse tion multipli ity of //f and Lo at 0

ideal L1 = y; vdim(std(f+L1)); //-> 3

//the spe ial line y=0 //interse tion multipli ity of //f and L1 at 0

When we want to ompute the number of interse tion points of ff = 0g with a small displa ement Le of Lo (where e is a small number), we have to be

areful: in the lo al ring r the polynomial de ning the displa ed line is a unit, hen e, the interse tion multipli ity is 0. Thus, we have to use a global ordering. However, ounting the interse tion number in a global ring gives the total interse tion number in aÆne spa e of ff = 0g with the given line (whi h is 3 for Lo as well as for Le ), and not only in a small Eu lidean neighbourhood of 0 (whi h is 2). The only thing we an do is to solve the system given by f = Le = 0 numeri ally and then \see", whi h points are lose to 0 (whi h is, in general, a guess). Let X be the parabola fx y 2 = 0g with mult(X; 0) = 1. The tangent

one is the y {axis, equal to L0 . A small displa ement of L0 interse ts X in more than mult(X; 0) points. The line L00 is transversal to the tangent one, L0" interse ts X in one point lose to 0 ( f. Figure A.21).

L0 L" 0 0

X

X

L0 L" Fig. A.21.

mult(X; 0) = 1; #X \ L" = 2; #X \ L" = 1. 0

What we dis ussed in the above examples was the interse tion multipli ity of a urve with a line. We generalize this to the interse tion multipli ity of two plane urves. In the following, K is again an algebrai ally losed eld (of arbitrary hara teristi ).

482


De nition A.8.16. (1) Let f; g 2 K [x; y ℄ be of positive degree, p = (p1 ; p2 ) 2 V (f ) \ V (g ) A 2K , and let mp = hx p1 ; y p2 i be the maximal ideal of p. We de ne Æ

i(f; g; p) := dimK K [x; y℄mp hf; gi ; and all it the interse tion multipli ity of f and g at p. (2) If C = Spe (K [x; y ℄=hf i) and D = Spe (K [x; y ℄=hg i), then

i(C; D; p) := i(f; g; p) is alled the interse tion multipli ity of C and D at p. (3) If C; D A 2K are lassi al aÆne plane urves and I (C ) = hf i, I (D) = hg i, then i(C; D; p) := i(f; g ; p) is alled the interse tion multipli ity of C and D at p. 5 Note that in (2) and (3) the generator of the (prin ipal) ideal hf i, respe tively hg i, is unique up to multipli ation by a non{zero onstant. Hen e, the interse tion multipli ity i(C; D; p) is well{de ned. Note also that i(f; g ; p) is nite if and only if f and g have no ommon fa tor whi h vanishes at p. This follows from Krull's prin ipal ideal theorem ( f. Corollary 5.6.10) and Corollary 5.3.17.

Now let F; G 2 K [z; x; y ℄ be homogeneous polynomials of positive degree, let p = (p0 : p1 : p2 ) 2 V (F ) \ V (G) P2K , and let mp = hp0 x p1 z; p0 y p2 z i be the homogeneous ideal of p. Assume that p0 6= 0, then it is easy to see that Æ

i(F; G; p) := dimK K [z; x; y℄mp hF; Gi = i(f; g; p) ; where f = F a = F jz=1 and g = Ga = Gjz=1 are the aÆnizations of F and G. i(F; G; p) is alled the interse tion multipli ity of F and G at p. Note that, by de nition, it is a purely lo al invariant. If C = Proj(K [z; x; y ℄=hf i) and D = Proj(K [z; x; y ℄=hg i), respe tively, if C; D P2K are lassi al proje tive plane urves with I (C ) = hF i, I (D) = hGi, then we set

i(C; D; p) := i(F; G; p) ; and all it the interse tion multipli ity of the proje tive plane urves C and

D at p.

It is lear that i(F; G; p) = 0 if and only if f or g is a unit in the lo al ring K [x; y℄hx p1 ;y p2 i , and the latter is equivalent to p 62 V (F ) \ V (G). Moreover, i(F; G; p) < 1 for all p 2 P2K if and only if F and G have no ommon 5 The dieren e here is just that f and g are not allowed to have multiple fa tors whi h are allowed in (1) and (2).


483

non- onstant fa tor. It is also lear that i(f r; g s ; p) i(f; g ; p) for r; s 1, hen e, i(Fred ; Gred ; p) i(F; G; p), where Fred ; Gred denote the squarefree parts of F; G, respe tively. Hen e, the s heme{theoreti interse tion multipli ity is at least the interse tion multipli ity of the redu ed urves. Now we an formulate one of the most important theorems in proje tive geometry, ombining lo al and global invariants:

Theorem A.8.17 (Bezout's Theorem). Let F; G 2 K [z; x; y℄ be polyno-

mials of positive degree, without ommon non{ onstant fa tor. Then the interse tion V (F ) \ V (G) is a nite set and deg(F ) deg(G) =

X

p2V (F )\V (G)

i(F; G; p) :

Sin e the right{hand side is zero if and only if V (F ) \ V (G) is empty, we obtain:

Corollary A.8.18. Let C; D 2 P2K be two proje tive plane urves without

ommon omponents, C = V (F ), D = V (G), then

deg(F ) deg(G) #(C \ D) 1 :

Proof of Theorem A.8.17. Sin e F and G have no ommon fa tor, F is a non{ zerodivisor in K [z; x; y ℄=hGi. By Lemma 5.3.5, deg(G) = deg(K [z; x; y ℄=hGi), and, by Proposition 5.3.6, we obtain

deg(K [z; x; y ℄=hF; Gi) = deg(F ) deg(G) :

By Krull's prin ipal ideal theorem 5.6.8, dim(K [z; x; y ℄=hF; Gi) = 1. Now Theorem 5.3.7 implies that the Hilbert polynomial of K [z; x; y ℄=hF; Gi is onstant and, by de nition, this onstant is deg(K [z; x; y ℄=hF; Gi). In parti ular, the proje tive variety V (F; G) P2K has dimension 0, and we may assume (after a linear hange of oordinates) that V (F; G) \ V (z ) = ; and F and G are not divisible by z . Let f = F a = F jz=1 and g = Ga = Gjz=1 2 K [x; y ℄ be the aÆnizations of F and G. Then, by Remark 5.3.16, the Hilbert fun tion of K [z; x; y℄=hF; Gi

oin ides with the aÆne Hilbert fun tion of K [x; y ℄=hf; g i and, hen e, deg(K [z; x; y ℄=hF; Gi) = dimK (K [x; y ℄=hf; g i) :

Let V (f ) \ V (g ) = fp1 ; : : : ; pr g, pi = (pi1 ; pi2 ), and let mi = hx pi1 ; y pi2 i be the orresponding maximal ideals, i = 1; : : : ; r. Then, by the Chinese remainder theorem ( f. Exer ise 1.3.13),

K [x; y℄=hf; gi =

r M i=1

K [x; y℄=Qi ;

where the Qi are mi {primary ideals. Now K [x; y ℄=Qi = K [x; y ℄mi =hf; g i, and the result follows.

484


A.9 Singularities

In Se tion A.8 we have already de ned singular points as points p of a variety X where the lo al ring OX;p is not regular. In parti ular, \singular" is a lo al notion, where \lo al" so far was mainly onsidered with respe t to the Zariski topology. However, sin e the Zariski topology is so oarse, small neighbourhoods in the Zariski topology might not be lo al enough. If our eld K is C , then we may use the Eu lidean topology and we an study singular points p in arbitrary small "{neighbourhoods (as we did already at the end of Se tion A.8). But then we must also allow more fun tions, sin e the regular fun tions at p (in the sense of De nition A.6.1) are always de ned in a Zariski neighbourhood of p. Thus, instead of onsidering germs of regular fun tions at p, we onsider germs of omplex analyti fun tions at p = (p1 ; : : : ; pn ) 2 C . The ring of these fun tions is isomorphi to the ring of onvergent power series C fx1 p1 ; : : : ; xn pn g, whi h is a lo al ring and

ontains the ring C [x1 ; : : : ; xn ℄hx1 p1 ;:::;xn pn i of regular fun tions at p. For arbitrary (algebrai ally losed) elds K , we annot talk about onvergen e and then a substitute for C fx1 p1 ; : : : ; xn pn g is the formal power series ring K [[x1 p1 ; : : : ; xn pn ℄℄. Unfortunately, with formal power series, we annot go into a neighbourhood of p; formal power series are just not de ned there. Therefore, when talking about geometry of singularities, we

onsider K = C and onvergent power series. Usually, the algebrai statements whi h hold for onvergent power series do also hold for formal power series (but are easier to prove sin e we need no onvergen e onsiderations). We just mention in passing that there is, for varieties over general elds, another notion of \lo al" with etale neighbourhoods and Henselian rings ( f. [115℄) whi h is a geometri substitute of onvergent power series over C . For I C [x℄, x = (x1 ; : : : ; xn ), an ideal, we have in lusions of rings

C [x℄=I C [x℄hxi =I C [x℄hxi C fxg=I C fxg C [[x℄℄=I C [[x℄℄ : To distinguish the dierent points of view, we make the following de nition:

De nition A.9.1. Let X C n be an algebrai variety and p 2 X . The an-

alyti germ (X; p) of X at p is an equivalen e lass of open neighbourhoods of p in X , in the Eu lidean topology, where any two open neighbourhoods of p in X are equivalent. The analyti lo al ring of X at p is the ring of germs of omplex analyti fun tions at p = (p1 ; : : : ; pn ), an := C fx OX;p 1 p1 ; : : : ; xn pn g=I (X ) C fx1 p1 ; : : : ; xn If X A nK , K an arbitrary algebrai ally losed eld, let OX;p = K [x1 ; : : : ; xn ℄hx1 p1 ;:::;xn pn i

pn g :

be the algebrai lo al ring of X at the ( losed) point p = (p1 ; : : : ; pn ) and

A.9 Singularities an := O bX;p = K [[x1 OX;p

485

p1; : : : ; xn pn ℄℄=I (X ) K [[x1 p1 ; : : : ; xn pn℄℄

be the analyti lo al ring of X at p, b denoting the hx1 p1 ; : : : ; xn pn i{adi

ompletion. an , also a We all the analyti germ (X; p), and the analyti lo al ring OX;p singularity . In the following, we write K hx1 ; : : : ; xn i to denote either K [[x1 ; : : : ; xn ℄℄ or C fx1 ; : : : ; xn g. For analyti singularities, regular (in the sense of non{singular) points have a very ni e interpretation. By the Ja obian riterion and the impli it fun tion theorem, p is a regular point of the omplex variety X if, in a small Eu lidean neighbourhood of p, X is a omplex manifold. Algebrai ally, an being isomorphi to K hx ; : : : ; x i, as analyti this is equivalent to OX;p 1 d algebra, whi h holds for onvergent and formal power series (but not for regular fun tions in the sense of De nition A.6.1). Many invariants of singularities, whi h are de ned in the onvergent, respe tively formal, power series ring, an be omputed in the lo alization C [x℄hxi but this is not always the ase, as the following shows. Consider the singularity at 0 = (0; 0) of the plane urve y 2 x2 (1 + x) = 0 ( f. Figure A.22).

Fig. A.22.

The variety V

y2 x2 (1 + x).

The pi ture shows that in a small neighbourhood of 0 (with respe t to the Eu lidean topology) the urve has two irredu ible omponents, meeting transversally, but in the aÆne plane the urve is irredu ible. To see this algebrai ally, let us onsider f = y 2 x2 (1 + x) as an element of C fx; y g . We have a de omposition

p

f= y

p

p

x 1+x y+x 1+x

with y x 1 + x 2 C fx; y g , that is, f an be fa torized in C fx; y g into two non{trivial fa tors. The zero{set of these fa tors orresponds to the two omponents of V (f ) in a small neighbourhood of 0. This is also the fa torization in C [[x; y℄℄, sin e the fa torization is unique. However, f is irredu ible in C [x; y℄,

486


and even in C [x; y ℄hx;yi . Otherwise, there would exist g; h 2 C [x; y ℄hx;yi satisfying f = (y + xg )(y + xh), hen e g = h and g 2 = 1 + x. But, sin e 1 + x is de ned everywhere, g 2 and, hen e, g must be a polynomial, whi h is impossible sin e g 2 has degree 1. We an imagine the irredu ibility of f in C [x; y℄hx;yi also geometri ally, sin e this orresponds to the irredu ibility of the urve ff = 0g in an arbitrary small neighbourhood of 0 with regard to the Zariski topology. Su h a neighbourhood onsists of the urve minus nitely many points dierent from 0. Sin e the urve is a omplex urve and sin e a

onne ted open subset of C minus nitely many points is irredu ible (here, the above real pi ture is misleading), we an \see" the irredu ibility of f in C [x; y℄hx;yi (this argument an a tually be turned into a proof). The above example shows that the Zariski neighbourhoods are too big for

ertain purposes, or, algebrai ally, the algebrai lo al rings K [x℄hxi =I K [x℄hxi are too small and we have to work with the analyti lo al rings , K hxi=IK hxi. The basi theorem for the study of C fxg and K [[x℄℄ is the Weierstra preparation theorem (whi h does not hold for K [x℄hxi ) and whi h is treated in Se tion 6.2 for formal power series. Computationally, however, we an, basi ally, treat only K [x℄ and K [x℄hxi or fa tor rings of those in Singular. In parti ular, we annot put a polynomial into Weierstra normal form, nor fa torize it in K [[x℄℄, ee tively in Singular. We are only able to do so approximately, up to a given order (not to mention the problem of oding an in nite, but not algebrai , power series). Nevertheless, it has turned out that many invariants of singularities an be omputed in K [x℄hxi , an algebrai reason for this being the fa t that K [x℄hxi K [[x℄℄ is faithfully at (Corollary 7.4.7). We illustrate this with a few examples, the simplest is stated in the following lemma:

Lemma A.9.2. Let OX;p be the algebrai lo al ring of a variety at p 2 X

and let I OX;p be an ideal su h that dimK (OX;p =I ) < 1. Then, as lo al an =I Oan . In parti ular, both ve tor spa es have the k{algebras, OX;p =I = OX;p X;p same dimension and a ommon basis represented by monomials. To see this, we may assume p = 0. Then hx1 ; : : : ; xn is I OX;0 for some s, by an =hx ; : : : ; x is . The same result assumption, and OX;0 =hx1 ; : : : ; xn is = OX; 1 n 0 r holds for submodules I OX;0 of nite K { odimension. However, if I OX;0 is generated by polynomials f1 ; : : : ; fk , then there may be no s su h that hx1 ; : : : ; xn is hf1 ; : : : ; fk i K [x1 ; : : : ; xn ℄ (this in lusion holds if and only if V (f1 ; : : : ; fk ) = f0g) and, therefore, dimK OX;0 =I = 6 dimK K [x1 ; : : : ; xn ℄=hf1 ; : : : ; fk i ; in general. Important examples are given by the Milnor number and by the Tjurina number of an isolated hypersurfa e singularity.

A.9 Singularities

487

De nition A.9.3. (1) We say that f 2 K [x℄, x = (x1 ; : : : ; xn ), has an isolated riti al point at p if p is an isolated point of V (f=x1; : : : ; f=xn). Similarly, we say that p is an isolated singularity of f , or of the hypersurfa e V (f ) A nK , if p is an isolated point of V (f; f=x1 ; : : : ; f=xn ). 6 (2) We all the number

(f; p) := dimK K hx1 p1 ; : : : ; xn pn i the Milnor number , and

(f; p) := dimK K hx1 p1 ; : : : ; xn pn i

f f ;:::; x1 xn

f;

f f ;:::; x1 xn

the Tjurina number of f at p. We write (f ) and (f ) if p = 0.

If p is an isolated riti al point of f , then mp := hx1 p1 ; : : : ; xn pn i is a minimal asso iated prime of hf=x1 ; : : : ; f=xn i by the Hilbert Nullstellensatz and, therefore, msp hf=x1 ; : : : ; f=xn i K [x℄mp for some s. It follows that the Milnor number (f; p) is nite and, similarly, if p is an isolated singularity of V (f ), then the Tjurina number (f; p) is nite, too. By Lemma A.9.2 we an ompute the Milnor number (f ), resp. the Tjurina number (f ), by omputing a standard basis of hf=x1 ; : : : ; f=xn i, respe tively hf; f=x1 ; : : : ; f=xn i with respe t to a lo al monomial ordering and then apply the Singular ommand vdim. We an use the interplay between lo al and global orderings to he k the existen e of riti al points and of singularities outside 0. For this we use the (easy) fa ts:

(f; p) = 0 if and only if p is a non{ riti al point of f , that is, p 62 V

f f =: Crit(f ) ; ;:::; x1 xn

(f; p) = 0 if and only if p is a non{singular point point of V (f ), that is,

f f p 62 V f; =: Sing(f ) : ;:::; x1 xn Note that we have the following equalities for the total Milnor number , respe tively the total Tjurina number , of f :

dimK K [x1 ; : : : ; xn ℄

f f ;:::; x1 xn

=

X

p2 Crit(f )

(f; p) ;

6 Examples of isolated and non{isolated hypersurfa e singularities are shown in Figure A.23.

488

A. Geometri Ba kground Isolated Singularities

A1 : x2 y2 + z2 = 0

D4 : z3 zx2 + y2 = 0

Non{isolated singularities

A : x2 y 2 = 0

D

1

Fig. A.23.

1

: y2

zx2 = 0

Isolated and non{isolated singularities.

dimK K [x1 ; : : : ; xn ℄

f;

f f ;:::; x1 xn

=

X

p2 Sing(f )

(f; p) ;

Moreover, note that a singular point of f is a riti al point of f whi h lies on the hypersurfa e ff = 0g.

SINGULAR Example A.9.4 (Milnor and Tjurina number).

We ompute the lo al and the total Milnor, respe tively Tjurina, number and he k in this way, whether there are further riti al, respe tively singular, points outside 0. LIB "sing.lib"; ring r = 0,(x,y,z),ds; //lo al ring poly f = x7+y7+(x-y)^2*x2y2+z2; milnor(f); //-> 28 //Milnor number at 0 tjurina(f); //-> 24 //Tjurina number at 0

A.9 Singularities

489

ring R = 0,(x,y,z),dp; //affine ring poly f = x7+y7+(x-y)^2*x2y2+z2; milnor(f); //-> 36 //total Milnor number tjurina(f); //-> 24 //total Tjurina number

We see that the dieren e between the total and the lo al Milnor number is 8; hen e, f has eight riti al points ( ounted with their respe tive Milnor numbers) outside 0. On the other hand, sin e the total Tjurina number oin ides with the lo al Tjurina number, V (f ) A 3 has no other singular points ex ept 0. The most simple singularities of a hypersurfa e are (ordinary) nodes: a riti al point p = (p1 ; : : : ; pn ) of f 2 K [x1 ; : : : ; xn ℄ is alled a node or an A1 { singularity if there exist analyti oordinates y1 ; : : : ; yn entred at p su h that f (y1 ; : : : ; yn ) = y12 + + yn2 (that is, there exist 'i 2 K [[y1 ; : : : ; yn ℄℄, 'i (0) = 0, i = 1; : : : ; n, su h that substituting xi by 'i + pi , i = 1; : : : ; n, indu es an isomorphism ' : K [[x1 p1 ; : : : ; xn pn ℄℄ ! K [[y1 ; : : : ; yn ℄℄ with '(f ) = y12 + + yn2 ). In this de nition, we assume that har(K ) 6= 2. By the Morse lemma ( f. [129℄, [87℄, [104℄), a riti al point p of f is a node if and only if the Hessian at p is nondegenerate, that is, if and only if

det

2f (p) = 6 0: xi xj

Moreover, as is not diÆ ult to see, this is also equivalent to (f; p) = 1 and also to (f f (p); p) = 1 (in hara teristi 0). Hen e, we an ount the number of nodes of a fun tion as in the following example:

SINGULAR Example A.9.5 ( ounting nodes).

Consider f from Example A.9.4 and use the rings de ned there. We ompute the ideal nn of riti al points whi h are not nodes. setring R; ideal j = ja ob(f); //ideal of riti al lo us poly h = det(ja ob(j)); //det of Hessian of f ideal nn = j,h; //ideal of non-nodes vdim(std(nn)); //-> 27 setring r; //go ba k to lo al ring ideal nn = ja ob(f),det(ja ob(ja ob(f))); vdim(std(nn)); //-> 27

The omputation in the aÆne ring R shows that there are, perhaps several, non{nodes. The ideal of non{nodes of f is generated by f=x1 ; : : : ; f=xn

490


and by the determinant of the Hessian det 2 f=(xi xj ) . The Singular

ommand vdim(std(nn)) ounts the number of non{nodes, ea h non{node p being ounted with the multipli ity

dimK K hx1 p1 ; : : : ; xn pn i

f f 2f ;:::; ; det x1 xn xi xj

(whi h is equal to or less than the lo al Milnor number, sin e the determinant of the Hessian redu es the multipli ity). The omputation in the lo al ring r shows, however, that 0 is the only non{node, sin e the multipli ity of the non{nodes is 27 in both ases. Hen e, all the riti al points in A 3 r f0g of f are nodes and there are 36 28 = 8 of them (the singularity at 0 is, of

ourse, not a node, sin e it has Milnor number 29). As the analyti lo al ring of a singularity is a quotient ring of a power series ring, singularity theory deals with power series rather than with polynomials. It is, however, a fundamental fa t that isolated hypersurfa e singularities are nitely determined by a suÆ iently high jet (power series expansion up to a suÆ iently high order). That is, if f 2 K hx1 ; : : : ; xn i has an isolated singularity at 0, then there exists a k > 0 su h that any g 2 K hx1 ; : : : ; xn i, having the same k{jet as f , is right equivalent to g (that is, there exists an automorphism ' of K hx1 ; : : : ; xn i su h that '(f ) = g ). We say that f is k{ determined in this situation, and the minimal su h k is alled the determina y of f . Hen e, if f has an isolated singularity, then we an repla e it by its k {jet (whi h is a polynomial), without hanging the singularity. The nite determina y has important onsequen es, theoreti al as well as omputational ones. A typi alPappli ation is the onstru tion of analyti

oordinate transformations ' = 1 ' of K hx1 ; : : : ; xn i, where ' is homogeneous of degree and whi h are onstru ted degree by degree. Usually, will not stop, but if f is k {determined, then we know that P this pro ess =1;:::;k ' (f ) is right equivalent to f . As a general estimate, we have that f 2 K hx1 ; : : : ; xn i is (f ) + 1 { determined, provided har(K ) = 0 ( f. [104℄, [87℄). Using this estimate, the Morse lemma ited above is an easy onsequen e: it follows that for f 2 hx1 ; : : : ; xn i2 and (f ) = 1, f is right equivalent to its 2{jet and, hen e, a node. The estimate (f ) + 1 for the determina y is, in general, not very good. Instead, we have a mu h better estimate given by the following theorem ( f. [104℄, [87℄):

Theorem A.9.6. Let f 2 m = hx1 ; : : : ; xn i K hx1 ; : : : ; xn i, where K is a eld of hara teristi 0. Then

mk+1 m2 implies that f is k{determined.

f f ;:::; x1 xn

A.9 Singularities

491

In parti ular, if mk m hf=x1 ; : : : ; f=xn i, then f is k {determined. Of

ourse, we an ompute, by the method of Se tion 1.8.1, the minimal k satisfying mk+1 I := m2 hf=x1 ; : : : ; f=xn i, by omputing a standard basis G of I and then a normal form for mi with respe t to G, for in reasing i. However, this an be avoided due to a very useful feature of Singular, the so{ alled highest orner ( f. De nition 1.7.11). The highest orner of an ideal I is the minimal monomial m (with respe t to the monomial ordering) su h that m 62 I . In ase of a lo al degree ordering the highest orner exists if and only if dimK K hx1 ; : : : ; xn i=I is nite (Lemma 1.7.14). If we ompute a standard basis of I , then Singular omputes automati ally the highest

orner, if it exists. The ommand high orner(I); returns it. Hen e, if we

ompute a standard basis of I with respe t to a lo al degree ordering and if the monomial m is the highest orner of I , then mdeg(m)+1 I . We obtain

Corollary A.9.7. Let har (K ) = 0, and let f 2 m K hx1 ; : : : ; xn i have an isolated singularity. Moreover, let mi 2 Mon(x1 ; : : : ; xn ) be the highest orner of mi hf=x1; : : : ; f=xni, i = 0; 1; 2 with respe t to a lo al degree ordering. Then f is deg(mi ) + 2 i determined.

SINGULAR Example A.9.8 (estimating the determina y).

We ompute the highest orner of hx; y ii hf=x; f=y i, i = 0; 1; 2, for the E7 {singularity f = x3 + xy3 2 C fx; yg and estimate the determina y. ring r = 0,(x,y),ds; poly f = x3+xy3; ideal j = ja ob(f); ideal j1 = maxideal(1)*j; ideal j2 = maxideal(2)*j; j = std(j); deg(high orner(j)); //-> 4 j1 = std(j1); deg(high orner(j1)); //-> 4 j2 = std(j2); deg(high orner(j2)); //-> 5

Corollary A.9.7 implies, using the ideal j, that f is 6{determined. If we use the ideals j1, respe tively j2, it follows that f is 5{determined. One an show that E7 is even 4{determined, hen e Corollary A.9.7 provides only an estimate for the determina y (whi h, nevertheless, is quite good in general). We should like to nish this se tion with two further appli ations of standard bases in lo al rings: lassi ation of singularities and deformation theory. There are many more, some of them are in the Singular libraries and we refer to the examples given there.

492


In a tremendous work, V.I. Arnold started, in the late sixties, the lassi ation of hypersurfa e singularities up to right equivalen e. His work ulminated in impressive lists of normal forms of singularities and, moreover, in a determinator for singularities whi h allows the determination of the normal form for a given power series ([3℄). This work of Arnold has found numerous appli ations in various areas of mathemati s, in luding singularity theory, algebrai geometry, dierential geometry, dierential equations, Lie group theory and theoreti al physi s. The work of Arnold was ontinued by many others, we just mention C.T.C. Wall [177℄. Most prominent is the list of ADE or simple or Kleinian singularities , whi h have appeared in surprisingly dierent areas of mathemati s, and still today, new onne tions of these singularities to other areas are being dis overed (see, for example, [48℄, [81℄). Here is the list of ADE{singularities for algebrai ally losed elds of hara teristi 0 (for the lassi ation in positive

hara teristi see [41℄). The names ome from their relation to the simple Lie groups of type A, D and E.

Ak : Dk : E6 : E7 : E8 :

xk1+1 + x22 + x23 + + x2n ; k 1; k 2 2 2 2 x1 (x1 + x2 ) + x3 + + xn ; k 4 ; x41 + x32 + x23 + + x2n ; x2 (x31 + x22 ) + x23 + + x2n ; x51 + x32 + x23 + + x2n :

A3 {singularity Fig. A.24.

D6 {singularity

E7 {singularity

Some two{dimensional simple singularities.

Arnold introdu ed the on ept of \modality", related to Riemann's idea of moduli, into singularity theory and lassi ed all singularities of modality 2 (and also of Milnor number 16), and many more. The ADE{singularities are just the singularities of modality 0. Singularities of modality 1 are the three paraboli singularities :

Ee6 = P8 = T333 : x3 + y3 + z 3 + axyz; a3 + 27 6= 0 ; Ee7 = X9 = T244 : x4 + y4 + ax2 y2 ; a2 6= 4 ; Ee8 = J10 = T236 : x3 + y5 + ax2 y2 ; 4a3 + 27 6= 0 ;

A.9 Singularities

493

the 3{indexed series of hyperboli singularities

Tpqr : xp + yq + z r + axyz; a 6= 0;

1

p

+

1

q

+

1

r

< 1;

and 14 ex eptional families, f. [3℄. The proof of Arnold for his determinator is, to a great part, onstru tive and has been partly implemented in Singular, f. [112℄. Although the whole theory and the proofs deal with power series, everything an be redu ed to polynomial omputations, sin e we deal with isolated singularities, whi h are nitely determined, as explained above. An important initial step in Arnold's lassi ation is the generalized Morse lemma , or splitting lemma , whi h says that

f Æ '(x1 ; : : : ; xn ) = x21 + + x2r + g(xr+1 ; : : : ; xn ) ; for some analyti oordinate hange ' and some power series g 2 m3 , if the rank of the Hessian matrix of f at 0 is r. The determina y allows the omputation of ' up to suÆ iently high order and the polynomial g . This has been implemented in Singular and is a

ornerstone in lassifying hypersurfa e singularities. In the following example we use Singular to obtain the singularity T5;7;11 from a database A L (\Arnold's list"), make some oordinate hange and determine then the normal form of the ompli ated polynomial after oordinate

hange.

SINGULAR Example A.9.9 ( lassi ation of singularities). LIB " lassify.lib"; ring r = 0,(x,y,z),ds; poly f = A_L("T[5,7,11℄"); f; //-> xyz+x5+y7+z11 map phi = r, x+z,y-y2,z-x; poly g = phi(f); g; //-> -x2y+yz2+x2y2-y2z2+x5+5x4z+10x3z2+10x2z3+5xz4+z5+y7 //-> -7y8+21y9-35y10-x11+35y11+11x10z-55x9z2+165x8z3 //-> -330x7z4+462x6z5-462x5z6+330x4z7-165x3z8+55x2z9 //-> -11xz10+z11-21y12+7y13-y14 qui k lass(g); //-> Singularity R-equivalent to : T[k,r,s℄=T[5,7,11℄ //-> normal form : xyz+x5+y7+z11 //-> xyz+x5+y7+z11

Beyond lassi ation by normal forms, the onstru tion of moduli spa es for singularities, for varieties or for ve tor bundles is a pretentious goal, theoreti ally as well as omputational. First steps towards this goal for singularities were undertaken in [15℄ and [61℄.

494


Let us nish with a few remarks about deformation theory . Consider a singularity (X; 0) given by power series f1 ; : : : ; fk 2 K hx1 ; : : : ; xn i. The idea of deformation theory is to perturb the de ning fun tions in a ontrolled way, that is, we onsider power series F1 (t; x); : : : ; Fk (t; x) with Fi (0; x) = fi (x), where t 2 S may be onsidered as a small parameter of a parameter spa e S ( ontaining 0). For t 2 S , the power series fi;t (x) = Fi (t; x) de ne a singularity Xt , whi h is a perturbation of X = X0 for t 6= 0 lose to 0. It may be hoped that Xt is simpler than X0 , but still ontains enough information about X0 . For this hope to be ful lled, it is, however, ne essary to restri t the possible perturbations of the equations to at perturbations, whi h are alled deformations ( f. Chapter 7 for the notion of atness). By a theorem of Grauert [79℄, for any isolated singularity (X; 0) there exists a semi{universal (or miniversal ) deformation , whi h ontains essentially all information about all deformations of (X; 0). For an isolated hypersurfa e singularity f (x1 ; : : : ; xn ), the semi{universal deformation is given by

F (t; x) = f (x) +

X j =1

tj gj (x) ;

where 1 =: g1 ; g2 ; : : : ; g represent a K {basis of the Tjurina algebra

K hx1 ; : : : ; xn i

f f f; ;:::; x1 xn

;

being the Tjurina number. To ompute g1 ; : : : ; g we only need to ompute a standard basis of the ideal hf; f=x1 ; : : : ; f=xn i with respe t to a lo al ordering and then ompute a basis of K [x℄ modulo the leading monomials of the standard basis.

For omplete interse tions, that is, singularities whose analyti lo al rings are omplete interse tion rings, we have similar formulas. For non{hypersurfa e singularities, the semi{universal deformation is mu h more ompli ated and, up to now, no nite algorithm is known in general. However, there exists an algorithm to ompute this deformation up to arbitrary high order ( f. [117℄, [124℄), whi h is implemented in Singular. As an example, we al ulate the semi{universal deformation of the normal surfa e singularity, being the one over the rational normal urve C of degree 4, parametrized by t 7! (t; t2 ; t3 ; t4 ). Homogeneous equations for the one over C are given by the 2 2{minors of the matrix:

x y z u m := y z u v

2 Mat(2 4; K [x; y; z; u; v℄) :

A.9 Singularities

Fig. A.25.

495

Deformation of an E7 {singularity in four A1 {singularities.

SINGULAR Example A.9.10 (deformation of singularities). LIB "deform.lib"; ring r = 0,(x,y,z,u,v),ds; matrix m[2℄[4℄ = x,y,z,u,y,z,u,v; ideal f = minor(m,2); //ideal of 2x2 minors of m versal(f); // omputes semi-universal deformation setring Px; //data are ontained in the ring Px Fs; //-> //-> //-> //-> //-> //-> Js; //-> //-> //->

Fs[1,1℄=-u2+zv+Bu+Dv Fs[1,2℄=-zu+yv-Au+Du Fs[1,3℄=-yu+xv+Cu+Dz Fs[1,4℄=z2-yu+Az+By Fs[1,5℄=yz-xu+Bx-Cz Fs[1,6℄=-y2+xz+Ax+Cy Js[1,1℄=BD Js[1,2℄=-AD+D2 Js[1,3℄=-CD

The ideal Js = hBD; AD D2 ; CDi K [A; B; C; D℄ de nes the required base spa e, whi h onsists of a 3{dimensional omponent (D = 0) and a transversal 1{dimensional omponent (B = C = A D = 0). This was the rst example, found by Pinkham, of a base spa e of a normal surfa e singularity having several omponents of dierent dimensions. The full versal deformation is given by the anoni al map (Fs and Js as above) K [[A; B; C; D℄℄=Js ! K [[A; B; C; D; x; y; z; u; v ℄℄=hJs ; Fs i. Although, in general, the equations for the versal deformation are formal power series, in many ases of interest (as in the example above) the algorithm terminates and the resulting ideals are polynomial. Finally, let us ompute the dis riminant of the semi-universal deformation of the hypersurfa e singularity A3 .

496

A. Geometri Ba kground ring s = 0,x,ds; poly f = x4; versal(f); setring Px; Fs; //-> Fs[1,1℄=x4+Ax2+Bx+C

//the semi-universal deformation

The dis riminant of Fs is the ideal des ribing the lo us of points (A; B; C ) in the base spa e of the semi{universal deformation su h that the polynomial x4 + Ax2 + Bx + C has multiple roots. We ompute it by proje ting the singular lo us of the semi{universal deformation to the parameter spa e. ideal sing = Fs,diff(Fs,x); //the singular lo us eliminate(sing,x); //the dis riminant //-> _[1℄=256C3-27B4+144AB2C-128A2C2-4A3B2+16A4C

Let us plot the dis riminant. LIB"surf.lib"; plot(256C3-27B4+144AB2C-128A2C2-4A3B2+16A4C);

Swallow tail , the dis riminant of the semi-universal deformation of the hypersurfa e singularity A3 .

Fig. A.26.

B. SINGULAR | A Short Introdu tion

De omputer is niet de steen, maar de slijpsteen der wijzen.

(The omputer is not the philosophers' stone but the philosophers' whetstone.)

Hugo Battus, Rekenen op taal (1983).

In this se tion we shall give a short introdu tion to the omputer algebra system Singular . In all the hapters of this book there are already many examples for the intera tive use of Singular and the possibility to write own programmes in the Singular programming language. For more details we refer to the Singular Manual, whi h is oered as online help for Singular , and an be found as a posts ript le in the distribution. See also the Singular examples in the distribution. We start with instru tions on how to obtain Singular , explain the rst steps on how to work with Singular and, nally, give an overview about the data types and fun tions of Singular .

B.1 Downloading Instru tions is available, free of harge, as a binary programme for most ommon hardware and software platforms. Release versions of Singular an be downloaded through ftp from our FTP site

Singular

ftp://www.mathematik.uni-kl.de/pub/Math/Singular/,

or, using your favourite WWW browser, from http://www.mathematik.uni-kl.de/ftp/pub/Math/Singular/.

To download

Singular

for an

RPM-based

Unix platform :1

Make sure that you have approximately 20 MByte of free disk spa e under /opt and follow these steps: 1 To install surf, you may either download surf--ix86-Linux.tar.gz from the en losed CD and extra t the les to /opt. Or you may download the sour e ode from http://sour eforge.net/proje ts/surf and ompile it, following the instru tions in the INSTALL le.

498


(1) There are the following Singular

RPM

pa kages:

Singular--1.i386.rpm | Singular (dynami ally linked) exe utables Singular and ESingular (requires shared libraries) Singular-stati --1.i386.rpm | Singular (stati ally linked) exe utables Singular and ESingular (does not require shared

Singular-share--1.noar h.rpm | Singular libraries,

Singular-redhat--1.noar h.rpm | Gnome and KDE

libraries)

do umentation, and examples support for RedHat Linux

Singular-suse--1.noar h.rpm | Gnome and KDE sup-

port for SuSE Linux (2) Choose the RPM pa kages you need and download them. (3) To install or deinstall RPM pa kages, you need to log in as a superuser (type su in a ommand shell and enter the root password). (4) As a superuser, you an install the Singular RPM pa kages using a graphi al RPM frontend like e.g. gnorpm under Gnome or kpa kage under KDE.2 Alternatively, you an use the RPM ommand line tool rpm. To install an RPM pa kage Singular-...--1.i386.rpm using rpm, type rpm -ivh Singular-...--1.i386.rpm in a

ommand shell.

To download

Singular

for a non

RPM-based

Unix platform :3

Make sure that you have approximately 20 MByte of free disk spa e and follow these steps (you don't need superuser privileges): (1) You need to download two (ar hive) les: Singular--share.tar.gz | ar hite ture independent data su h as do umentation and libraries Singular--.tar.gz | ar hite ture dependent exe utables, su h as the Singular programme Here, an, for example, be one of the following: ix86-Linux | PC's running under Linux with lib version 6 (e.g. SuSE Linux distribution version 6 or RedHat Linux distribution version 5:2) ix86-Linux-lib 5 | PC's running under Linux with lib version 5 (run ldd /bin/ls to nd out your lib version) HPUX-9 | HP workstations running under HPUX version 9 HPUX-10 | HP workstations running under HPUX version 10 2 Under Gnome or KDE, simply double- li k on the i on of a Singular RPM pa kage to start the standard RPM appli ation. 3 To install surf, you may either download surf--ix86-Linux.tar.gz from the CD and extra t the les to the dire tory where you installed Singular. Or download the sour e ode from http://sour eforge.net/proje ts/surf and ompile it, following the instru tions in the INSTALL le.

B.1 Downloading Instru tions

499

SunOS-5 | Sun workstations running Solaris version 5 (2) Simply hange to the dire tory into whi h you wish to install Singular (usually wherever you install third{party software):

d /usr/lo al/

spe i subdire tories will be reated in su h a way that multiple versions and multiple ar hite ture dependent les of Singular an pea eably oexist under the same /usr/lo al/ tree. (3) Unpa k the ar hives: Singular

gzip -d Singular--.tar.gz | tar -pxf gzip -d Singular--share.tar.gz | tar -pxf -

This reates the following (sub)dire tories, whi h ontain Singular// Singular and ESingular programmes Singular//LIB Singular libraries (*.lib les) Singular//ema s Singular Ema s user interfa e Singular//info info les of Singular manual Singular//html html les of Singular manual Singular//do mis ellaneous do umentation les Singular//examples Singular examples (*.sing les)

To download

Singular

for Windows 95/98/ME/NT/2000/XP :

(1) Download one of the following self{extra ting ar hives: Singular--Compa t.exe

Minimal ar hive to download. Installs Singular , minimal set of needed tools and the Singular Manual in WinHelp format.

Singular--Typi al.exe

Typi al ar hive to download. Installs Singular with ne essary tools, the XEma s editor, ESingular and the Singular Manual in WinHelp and Html formats. Double- li k (or exe ute) the self{extra ting ar hives, and arefully follow the instru tions given there.

To download

Singular

for the Ma intosh :

There are two dierent possibilities: (1) For Ma OS X take =pp Ma -darwin and follow the instru tions for a Unix platform. (2) For older Ma OS systems Singular is an MPW{tool and the last distribution for su h systems. Make sure that you have MPW installed on your system. Download the le Singular--pp -MPW.sea-hqx. Expand the downloaded le using a standard expander to obtain the orresponding self{extra ting Singular--pp -MPW.sea le. Extra t this le. This will reate

500


a new folder named \Singular:" whi h ontains the `Singular' programme in the subfolder \:pp -MPW:". Do not hange the stru ture of the subfolders. Make the folder whi h ontains the exe utable the working dire tory of the MPW shell. To start Singular, you need to type in `Singular' in the MPW worksheet.

B.2 Getting Started

an either be run in text terminal or within Ema s. To start Singular in its text terminal user interfa e, enter Singular (or Singular-) at the system prompt. The Singular banner appears whi h, among others, reports the version (and the ompilation date, sear h path, lo ation of programme and libraries, et .): Singular

SINGULAR A Computer Algebra System for Polynomial Computations

/ / version 0< by: G.-M. Greuel, G. Pfister, H. S hoenemann \ April 2002 FB Mathematik der Universitaet, D-67653 Kaiserslautern \

To start Singular in its Ema s user interfa e enter ESingular at the system prompt. Generally, we re ommend using Singular in its Ema s interfa e, sin e this oers many more features and is more onvenient to use than the ASCII{ terminal interfa e (in parti ular, if running under Windows). To exit Singular type quit;, exit; or $ (or, when running within Ema s preferably type CTRL-C $). To exit during omputation type CTRL-C. There are a few important notes whi h one should keep in mind:

Every ommand has to be terminated by a ; (semi olon) followed by a Return . The online help is a

essible by means of help; or ?;.

Singular is a spe ial purpose system for polynomial omputations. Hen e, most of the powerful omputations in Singular require the prior de nition of a ring. The most important rings are polynomial rings over a eld, lo alizations thereof, or quotient rings of su h rings modulo an ideal. However, some simple omputations with integers (ma hine integers of limited size) and manipulations of strings are available without a ring. On e Singular is started, it awaits an input after the prompt >. Every statement has to be terminated by ; . If you type 37+5; followed by a Return, then 42 will appear on the s reen. We always start the Singular output with //-> to have it learly separated from the input.

37+5; //-> 42

B.2 Getting Started

501

All obje ts have a type, for example, integer variables are de ned by the word

int. An assignment is done by the symbol = . int k = 2;

Testing for equality, respe tively inequality, is done using ==, respe tively !=, (or ), where 0 represents the boolean value FALSE, any other integer value represents TRUE. k == //-> k != //->

2; 1 2; 0

The value of an obje t is displayed by simply typing its name. k; //-> 2

On the other hand, the output is suppressed if an assignment is made. int j; j = k+1;

The last displayed (!) result is always available with the spe ial symbol _. 2*_; //-> 4

// two times the value of k displayed above

Starting with // denotes a omment and the rest of the line is ignored in al ulations, as seen in the previous example. Furthermore, Singular maintains a history of the previous lines of input, whi h may be a

essed by CTRL-P (previous) and CTRL-N (next) or the arrows on the keyboard. To edit Singular input we have the following possibilities: TAB: CTRL-B: CTRL-F: CTRL-D: CTRL-P: RETURN:

automati name ompletion. moves the ursor to the left. moves the ursor to the right. deletes the symbol in the ursor. gives the pre eding line in the history. sends the urrent line to the Singular {Parser.

For more ommands see the Singular Manual. The Singular Manual is available online by typing the ommand help;. Explanations on single topi s, for example, on intmat, whi h de nes a matrix of integers, are obtained by help intmat;

Next, we de ne a 3 3 matrix of integers and initialize it with some values, row by row from left to right:

502

B. SINGULAR | A Short Introdu tion intmat m[3℄[3℄ = 1,2,3,4,5,6,7,8,9;

A single matrix entry may be sele ted and hanged using square bra kets [ and ℄. m[1,2℄=0; m; //-> 1,0,3, //-> 4,5,6, //-> 7,8,9

To al ulate the tra e of this matrix, we use a for loop. The urly bra kets {, respe tively }, denote the beginning, respe tively end, of a blo k. If you de ne a variable without giving an initial value, as the variable tr in the example below, Singular assigns a default value for the spe i type. In this ase, the default value for integers is 0. Note that the integer variable j has already been de ned above. int tr; for ( j=1; j 15 j++ is equivalent to j=j+1. To ount the even and the odd entries of the se ond row of this matrix, we use the if and else and a while loop. j=0; int even,odd; while(j 2 odd; //-> 1

Variables of type string an also be de ned and used without a ring being a tive. Strings are delimited by " (quotation{marks). They may be used to

omment the output of a omputation or to give it a ni e format. If a string

ontains valid Singular ommands, it an be exe uted using the fun tion exe ute. The result is the same as if the ommands were written on the

ommand line. This feature is espe ially useful to de ne new rings inside pro edures. "example for strings:"; //-> example for strings:

B.2 Getting Started

503

string s="The element of m "; s = s + "at position [2,3℄ is:"; //+ on atenates strings s , m[2,3℄ , "."; //-> The element of m at position [2,3℄ is: 6. s="m[2,1℄=0; m;"; exe ute(s); //-> 1,0,3, //-> 0,5,6, //-> 7,8,9

This example shows that expressions an be separated by , ( omma) giving a list of expressions. Singular evaluates ea h expression in this list and prints all results separated by spa es. To read data from a le, or to write it to a le, we an use the ommands read and write. For example, the following Singular session reates a le hallo.txt and writes 2 and example to this le. int a = 2; write("hallo.txt",a,"example");

This data an be read from the le again (as strings): read("hallo.txt"); //-> 2 //-> example

If we want to exe ute ommands from a le, we have to write exe ute(read("hallo.txt"));

or, in short, 1

To al ulate with obje ts as polynomials, ideals, matri es, modules, and polynomial ve tors, a ring has to be de ned rst: ring r = 0,(x,y,z),lp;

The de nition of a ring onsists of three parts: the rst part determines the ground eld, the se ond part determines the names of the ring variables, and the third part determines the monomial ordering to be used. So the example above de lares a polynomial ring alled r with a ground eld of hara teristi 0 (that is, the rational numbers) and ring variables alled x, y, and z. The lp at the end means that the lexi ographi al ordering is used.

504


The default ring in Singular is Z=32003[x; y; z ℄ with degree reverse lexi ographi al ordering: ring s; //-> //-> //-> //-> //->

s; // // // // //

hara teristi : 32003 number of vars : 3 blo k 1 : ordering dp : names x y z blo k 2 : ordering C

De ning a ring makes this ring the urrent a tive basering, so ea h ring de nition above swit hes to a new basering. The basering is now s. If we want to al ulate in the ring r, whi h we de ned rst, we have to swit h ba k to it. This an be done using the ommand setring: setring r;

On e a ring is a tive, we an de ne polynomials. A monomial, say x3 , may be entered in two ways: either using the power operator ^, saying x^3, or in shorthand notation without operator, saying x3. Note that the shorthand notation is forbidden if the name of the ring variable onsists of more than one hara ter. Note also that Singular always expands bra kets and automati ally sorts the terms with respe t to the monomial ordering of the basering. poly f = x3+y3+(x-y)*x2y2+z2; f; //-> x3y2-x2y3+x3+y3+z2

B.3 Pro edures and Libraries Singular oers a omfortable programming language, with a syntax lose to C. So it is possible to de ne pro edures whi h ombine several ommands to form a new one. Pro edures are de ned with the keyword pro followed by a name and an optional parameter list with spe i ed types. Finally, a pro edure may return values using the ommand return. Assume we want to ompute for f = x3 y 2 x2 y 3 + x3 + y 3 + z 2 2 Q [x; y; z ℄ the ve tor spa e dimension dimQ (Q [x; y; z ℄=hf=x; f=y; f=z i), the so{

alled global Milnor number of f (see also Page 487). Then we type

ring R=0,(x,y,z),dp; poly f=x3y2-x2y3+x3+y3+z2; ideal J=diff(f,x),diff(f,y),diff(f,z);

or

B.3 Pro edures and Libraries

505

ideal J=ja ob(f);

Then we have to ompute a standard basis of J : ideal K=std(J);

We obtain the Milnor number as vdim(K); //-> 12.

As a pro edure, we an write pro Milnor (poly h) { ideal J=ja ob(h); ideal K=std(J); int d=vdim(K); return(d); }

or, in a more ompa t form: pro Milnor (poly h) { return(vdim(std(ja ob(h)))); }

Note: if you have entered the rst line of the pro edure and pressed RETURN, prints the prompt . (dot) instead of the usual prompt > . This shows that the input is in omplete and Singular expe ts more lines. After typing the losing urly bra ket, Singular prints the usual prompt, indi ating that the input is now omplete. Then all the pro edure:

Singular

Milnor(f); //-> 12

If we want to ompute the lo al Milnor number of f at 0 ( f. De nition A.9.3), then we an use the same pro edure, but we have to de ne a lo al monomial ordering on R, for example, ring R=0,(x,y,z),ds;, and then de ne f; J; K , as above. The distribution of Singular ontains several libraries, whi h extend the fun tionality of Singular. Ea h of these libraries is a olle tion of useful pro edures based on the kernel ommands. The ommand help all.lib; lists all libraries together with a one{line explanation. One of these libraries is sing.lib whi h already ontains a pro edure

alled milnor to al ulate the Milnor number not only for hypersurfa es but, more generally, for omplete interse tion singularities. Libraries are loaded with the ommand LIB. Some additional information during the pro ess of loading is displayed on the s reen, whi h we omit here.

506

B. SINGULAR | A Short Introdu tion LIB "sing.lib";

As all input in Singular is ase sensitive, there is no on i t with the previously de ned pro edure Milnor, but the result is the same. milnor(f); //-> 12

Let us ompute the lo al Milnor number of f . We use the ommand imap to map f from R to the new basering with a lo al ordering: ring S=0,(x,y,z),ds; poly f=imap(R,f); milnor(f); //-> 4

(Sin e the global Milnor number is the sum of all lo al Milnor numbers at all riti al points of f , this shows that f must have further riti al points outside 0.) The pro edures in a library have a help part, whi h is displayed by typing help milnor;

as well as some examples, whi h are exe uted by example milnor;

Likewise, the library itself has a help part, to show a list of all the fun tions available for the user, whi h are ontained in the library. help sing.lib;

The output of the help ommands is omitted here.

B.4 Data Types In this se tion, we give an overview of all data types in Singular. For more details see the Manual.

def: Obje ts may be de ned without a spe i type: they get their type from

the rst assignment to them. For example, ideal i=x,y,z; def j=i^2; de nes the ideal i^2 with the name j. ideal: Ideals are represented as lists of polynomials, whi h generate the ideal: ideal I=x2-1,xy;. Like polynomials, they an only be de ned or a

essed with respe t to a basering.

ideal operations are:

+ addition ( on atenation of the generators and simpli ation), * multipli ation (with ideal, poly, ve tor, module; simpli ation in ase

of multipli ation with ideal),

^ exponentiation (by a nonnegative integer).

B.4 Data Types

507

ideal related fun tions are:

har_series, oeffs, ontra t, diff, degree, dim, eliminate, fa std, fa torize, fglm, finduni, groebner, high orner, homog, hilb, indepSet, interred, interse t, ja ob, jet, kbase, koszul, lead, lift, liftstd, lres, maxideal, minbase, minor, modulo, mres, mstd, mult, n ols, preimage, qhweight, quotient, redu e, res, simplify, size, sortve , sres, std, stdfglm, stdhilb, subst, syz, vdim, weight

int: Variables of type int represent the ma hine integers and are, therefore, limited in their range (for example, the range is between 2147483647 and 2147483647 on 32{bit ma hines): int i=1; .

int operations are:

hanges its operand to its su

essor, is itself not an int expression - hanges its operand to its prede essor, is itself not an int expression + addition negation or subtra tion * multipli ation / integer division (omitting the remainder), rounding toward 0 div integer division (omitting the remainder >=0) integer modulo (the remainder of the division /) % mod integer modulo (the remainder of the division div), always nonnegative ^, ** exponentiation (exponent must be nonnegative) , =, ==,

omparison An assignment j=i++; or j=i--; is not allowed, in parti ular it does not

hange the value of j. ++

int related fun tions are:

har, deg, det, dim, extg d, find, g d, koszul, memory, mult, n ols, npars, nrows, nvars, ord, par, pardeg, prime, random, regularity, rvar, size, tra e, var, vdim.

boolean expressions are:

A boolean expression is really an int expression used in a logi al ontext: an int expression ( 0 evaluates to TRUE (represented by 1), 0 represents FALSE ).

boolean operations are:

and (logi al), may also be written as &&, or (logi al), may also be written as ||, not (not logi al), may also be written as !

intmat: Integer matri es are matri es with integer entries. Integer matri es do not belong to a ring, they may be de ned without a basering being de ned: intmat im[2℄[3℄=1,2,3,4,5,6;.

508


intmat operations are:

+

* div,/ %, mod , ==

addition with intmat or int; the int is onverted into a diagonal intmat negation or subtra tion with intmat or int; the int is onverted into a diagonal intmat multipli ation with intmat, intve , or int; the int is onverted into a diagonal intmat division of entries in the integers (omitting the remainder) entries modulo int (remainder of the division)

omparison

intmat related fun tions are:

betti, det, n ols, nrows, random, size, transpose, tra e. iv=1,2,3,4;.

intve : Variables of type intve are lists of integers: intve intve operations are:

addition with intve or int ( omponent{wise)

hange of sign or subtra tion with intve or int ( omponent{ wise) * multipli ation with int ( omponent{wise) /, div division by int ( omponent{wise) %, mod modulo ( omponent{wise) , ==, =, >,
);

If no target is spe i ed, the result is printed. In some ases (for example,

export, keepring, kill, setring, type) the bra kets en losing the arguments are optional. For the ommands help, ontinue, break, quit and exit bra kets are not allowed.

attrib:

attrib ( name ) displays the attribute list of the obje t alled name. attrib ( name , string ) returns the value of the attribute string of the variable name. If the attribute is not de ned for this variable, attrib returns the empty string. attrib ( name , string , expression ) sets the attribute string of the variable name to the value expression. An attribute may be des ribed by any string. Some of these are used by the kernel of Singular and referred to as reserved attributes. Reserved attributes are: isSB, isHomog, isCI, isCM, rank, withSB, withHilb, withRes, withDim, withMult. bareiss: bareiss(M) applies the sparse Gau-Bareiss algorithm to a module (or with type onversion to a matrix) M using olumn operations with an \optimal" pivot strategy. It returns a module orresponding to a lower triangular matrix and a ve tor of integers giving the permutation of the rows made during the algorithm. Sin e no division by pivot elements takes pla e, it works over K [x1 ; : : : ; xn ℄, for example, for omputing the determinant.

B.5 Fun tions

betti:

513

betti(L) omputes the graded Betti numbers of Rn =M (R the baser-

ing, M a homogeneous submodule of Rn ) from the given resolution L of Rn =M .

har: har(R) returns the hara teristi of the oeÆ ient eld of the ring R.

har series: har_series(I) omputes a matrix. The rows of the matrix represent the irredu ible hara teristi series ( f. [132℄) of the ideal I with respe t to the urrent ordering of variables.

harstr: harstr(R) returns the des ription of the oeÆ ient eld of the ring R as a string.

leardenom: leardenom(f) multiplies the polynomial f , respe tively ve tor f , by a suitable onstant to an el all denominators of the oeÆ ients and then divides it by its ontent.

lose: lose(L) loses the link L.

oef: oef(f,m) determines the monomials in f divisible by one of the variables appearing in m (whi h is a produ t of ring variables) and the

oeÆ ients of these monomials as polynomials in the remaining variables.

oes: oeffs(J,z) develops ea h polynomial of J as a univariate polynomial in the given ring variable z , and returns the oeÆ ients as a k d matrix M , where d 1 is the maximal z {degree of all o

urring polynomials; k is the number of generators if J is an ideal (k = 1, if J is a polynomial). If J is a ve tor or a module this pro edure is repeated for ea h omponent and the resulting matri es are appended. An (optional) third argument T is used to return the matrix T of oeÆ ients su h that matrix(J) = T M .

oeffs(M,K,p) returns a matrix A of oeÆ ients with KA = M su h that the entries of A do not ontain any variable from p. Here K is a set of monomials, respe tively ve tors, with monomial entries, in the variables appearing in p, p is a produ t of variables (if this argument is not given, then the produ t of all ring variables is taken as default argument). M is supposed to onsist of elements of (respe tively have entries in) a nitely generated module over a ring in the variables not appearing in p. K should ontain the generators of M over this smaller ring. If K does not ontain all M , then KA = M 0 where M 0 is the part of M

orresponding to the monomials of K .

ontra t: ontra t(I,J) ontra ts ea h of the n elements of the se ond ideal J by ea h of the m elements of the rst ideal I , produ ing an m n{ matrix. Contra tion is de ned on monomials by: (

ontra t(xA ; xB ) :=

xB 0;

A;

if B A omponent{wise otherwise

where A and B are the multi{exponents of the ring variables represented by x. ontra t is extended bilinearly to all polynomials. de ned: defined(name) returns a value 0 (TRUE) if there is a user{ de ned obje t with this name, and 0 (FALSE) otherwise.

514

deg:


deg(f) returns the maximal (weighted) degree of the terms of a polynomial or a ve tor f ; the weights are the weights used for the rst blo k of the ring ordering. deg(0) is 1. degree: degree(I) omputes the (weighted) degree of the proje tive variety, respe tively sheaf over the proje tive variety, de ned by the ideal, respe tively module, generated by the leading monomials of the input. This is equal to the (weighted) degree of the proje tive variety, respe tively sheaf over the proje tive variety, de ned by the ideal, respe tively module, if the input is a standard basis with respe t to a (weighted) degree ordering. delete: delete(L,n) deletes the n{th element from the list L. di: diff(f,x) omputes the partial derivative of the polynomial f by the ring variable x. dim: dim(I) omputes the dimension of the ideal, respe tively module, generated by the leading monomials of the given generators of the ideal I , respe tively module. This is also the dimension of the ideal if it is represented by a standard basis. division: omputes a division with remainder. division(I,J) returns a list T,R,U where T is a matrix, R is an ideal, respe tively a module, and U is a diagonal matrix of units su h that matrix(I)*U=matrix(J)*T+matrix(R) is a standard representation for the (weak) normal form R of I with respe t to a standard basis of J. dump: dump(L) dumps (that is, writes in one \message" or \blo k") the state of the Singular session (that is, all de ned variables and their values) to the link L (whi h must be either an ASCII or MP link) su h that a getdump an retrieve it later on. eliminate: eliminate(I,m) eliminates variables o

urring as fa tors of the se ond argument m from an ideal I , respe tively module, by interse ting it with the subring not ontaining these variables. eliminate does not need a spe ial ordering nor a standard basis as input. Sin e elimination is expensive, for homogeneous input it might be useful rst to ompute the Hilbert fun tion of the ideal ( rst argument) with a fast ordering (for example, dp). Then make use of it to speed up the

omputation: a Hilbert{driven elimination uses the intve provided as the third argument. eval: evaluates (quoted) expressions. Within a quoted expression, the quote

an be \undone" by an eval (that is, ea h eval \undoes" the ee t of exa tly one quote). Used only when re eiving a quoted expression from an MP le link, with quote and write to prevent lo al evaluations when writing to an MPt p link. ERROR: immediately interrupts the urrent omputation, returns to the top level, and displays the argument string_expression as error message. This should be used as an emergen y, respe tively failure, exit within pro edures.

B.5 Fun tions

example:

515

example topi ; omputes an example for topi . Examples are available for all Singular kernel and library fun tions. exe ute: exe utes a string ontaining a sequen e of Singular ommands. exit: exits (quits) Singular , works also from inside a pro edure or from an interrupt. extg d: omputes extended g d: the rst element is the greatest ommon divisor of the two arguments, the se ond and third are fa tors su h that if list L=extg d(a,b); then L[1℄=a*L[2℄+b*L[3℄. Polynomials must be univariate to apply extg d. fa std: fa std(I) returns a list of ideals omputed by the fa torizing Grobner basis algorithm. The interse tion of these ideals has the same zero{set as the ideal I , that is, the radi al of the interse tion oin ides with the radi al of the input ideal. In many (but not all!) ases this is already a de omposition of the radi al of the ideal. (Note, however, that, in general, no in lusion between the input and output ideals holds.) A se ond, optional, argument an be a list of polynomials, whi h de ne non{zero onstraints. Hen e, the interse tion of the output ideal has a zero{set, whi h is the ( losure of the) omplement of the zero{set of the se ond argument in the zero{set of the rst argument. fa torize: fa torize(f) omputes the irredu ible fa tors (as an ideal) of the polynomial f together with or without the multipli ities (as an intve ) depending on the optional se ond argument: 0: returns fa tors and multipli ities, rst fa tor is a onstant. (fa torize(f) is a short notation for fa torize(f,0)). 1: returns non{ onstant fa tors (no multipli ities). 2: returns non{ onstant fa tors and multipli ities. fet h: fet h(R,I) maps the obje t I de ned over the ring R to the basering. fet h is the anoni al map between rings and qrings: the i{th variable of the sour e ring R is mapped to the i{th variable of the basering. The

oeÆ ient elds must be ompatible. Compared to imap, fet h uses the position of the ring variables, not their names. fglm: fglm(R,I) omputes for the ideal I in the ring R a redu ed Grobner basis in the basering, by applying the so{ alled FGLM (Faugere, Gianni, Lazard, Mora) algorithm. The main appli ation is to ompute a lexi ographi al Grobner basis from a redu ed Grobner basis with respe t to a degree ordering. This an be mu h faster than omputing a lexi ographi al Grobner basis dire tly. The ideal must be zero{dimensional and given as a redu ed Grobner basis in the ring R. fglmquot: fglmquot(I,p) omputes a redu ed Grobner basis of the ideal quotient I : hpi of a zero{dimensional ideal I and an ideal generated by a polynomial p, by using FGLM{te hniques. The ideal must be zero{ dimensional and given as a redu ed Grobner basis in the given ring. The polynomial must be redu ed with respe t to the ideal.

516


nd: returns the rst position of a substring in a string or 0 (if not found),

starts the sear h at the position given in the (optional) third argument. finduni(I) returns an ideal, whi h is ontained in the given ideal I su h that the i{th generator is a univariate polynomial in the i{th ring variable. The polynomials have minimal degree with respe t to this property. The ideal must be zero{dimensional and given as a redu ed Grobner basis in the urrent ring. freemodule: freemodule(n) reates the free module of rank n generated by gen(1),...,gen(n). g d: omputes the greatest ommon divisor of two integers or two polynomials. gen: gen(i) is the i{th free generator of a free module. getdump: getdump(L) reads the ontent of the entire le, respe tively link, L and restores all variables from it. For ASCII links, getdump is equivalent to an exe ute(read( link )) ommand. For MP links, getdump should only be used on data, whi h were previously dumped. groebner: groebner(I) omputes the standard basis of the argument I (ideal or module), by a heuristi ally hosen method: if the ordering of the

urrent ring is a lo al ordering, or if it is a non{blo k ordering and the

urrent ring has no parameters, then std(I) is returned. Otherwise, I is mapped into a ring with no parameters and ordering dp, where its Hilbert series is omputed. This is followed by a Hilbert series based standard basis

omputation in the original ring. If a se ond argument wait is given, then the omputation pro eeds at most wait se onds. That is, if no result ould be omputed in wait se onds, then the omputation is interrupted, 0 is returned, a warning message is displayed, and the global variable groebner_error is de ned. help: help topi ; displays online help information for topi using the

urrently set help browser. If no topi is given, the title page of the manual is displayed. ? may be used instead of help. topi an be an index entry of the Singular manual or the name of a (loaded) pro edure, whi h has a help se tion. topi may ontain wild ard hara ters (that is, * hara ters). If a (possibly \wild arded") topi annot be found (or uniquely mat hed) a warning is displayed and no help information is provided. If topi is the name of a (loaded) pro edure whose help se tion has

hanged w.r.t. the help available in the manual then, instead of displaying the respe tive help se tion of the manual in the help browser, the \newer" help se tion of the pro edure is simply printed to the terminal. The browser in whi h the help information is displayed an be set either with the ommand{line option

nduni:

--browser=

or, if

Singular

is already running, with the ommand

B.5 Fun tions

517

system("--browser", "").

Use the ommand

system("browsers");

for a list of all available browsers. high orner(I) returns the smallest monomial not ontained in the ideal, respe tively module, generated by the initial terms of the given generators of I . If the generators are a standard basis, this is also the smallest monomial not ontained in the ideal, respe tively module I . If the ideal, respe tively module, is not zero{dimensional, 0 is returned. Hen e, high orner is always 1 or 0 for global monomial orderings. hilb: omputes the (weighted) Hilbert series of the ideal, respe tively module, de ned by the leading terms of the generators of the given ideal, respe tively module. If hilb(I) is alled with one argument (the ideal or module I ), then the rst and se ond Hilbert series, together with some additional information, are displayed. If hilb(I,n) is alled with two arguments, then the n{th Hilbert series is returned as an intve , where n = 1; 2 is the se ond argument. More pre isely, if hilb(I,n) P = v0 ; : : : ; ; vd ; 0, then the n{th Hilbert series (Q(t), respe tively G(t)) is di=0 vi ti , f. page 278. If a weight ve tor w is a given as third argument, then the Hilbert series is omputed with respe t to these weights w (by default all weights are set to 1). The last entry of the returned intve is not part of the a tual Hilbert series, but is used in the Hilbert driven standard basis omputation. If the input is homogeneous with respe t to the weights and a standard basis, the result is the (weighted) Hilbert series of the original ideal, respe tively module. homog: homog(I) tests for homogeneity: returns 1 for homogeneous input, 0 otherwise. homog(I,t) homogenizes polynomials, ve tors, generators of ideals or modules I by multiplying ea h monomial with a suitable power of the given ring variable t (whi h must have weight 1). hres: omputes a free resolution of a homogeneous ideal using the Hilbert driven algorithm. More pre isely, let R be the basering and I be the given ideal. Then hres(I,k) omputes a minimal free resolution of R=I

high orner:

: : : ! F2

! F1 A!1 R ! R=I ! 0 :

A2

If k is not zero then the omputation stops after k steps and returns a list of modules Mi =module(Ai); i = 1 : : : k . list L=hres(I,0); returns a list L of n modules (where n is the number of variables of the basering) su h that L[i℄ = Mi in the above notation. imap: is the map between rings and qrings with ompatible ground elds, whi h is the identity on variables and parameters of the same name and 0 otherwise. imap(R,i) maps I de ned over the ring R to the basering. Compared with fet h, imap uses the names of variables and parameters. Unlike map and fet h, imap an map parameters to variables.

518


impart: returns the imaginary part of a number in a omplex ground eld, returns 0 otherwise.

indepSet:

indepSet(I) omputes a maximal set U of independent variables of the ideal I given by a standard basis. If v is the result then v [i℄ is 1 if and only if the i{th variable of the ring, x(i), is an independent variable. Hen e, the set U onsisting of all variables x(i) with v [i℄ = 1 is a maximal independent set. insert: insert(L,I) inserts a new element I into a list L at the rst pla e or (if alled with three arguments) after the given position. interred: interred(I) interredu es a set of polynomials or ve tors I . interse t: interse t(I,J) omputes the interse tion of the ideals, respe tively modules, I and J . ja ob: ja ob(f) omputes the Ja obian ideal, respe tively Ja obian matrix, generated by all partial derivatives of the input f . jet: jet(f,k) deletes from the rst argument, f , all terms of degree larger than the se ond argument, k . If a third argument, w, of type intve is given, the degree is repla ed by the weighted degree de ned by w. jet is independent of the given monomial ordering. kbase: kbase(I) omputes a ve tor spa e basis ( onsisting of monomials) of the quotient ring by the ideal, respe tively of a free module by the module, I , in ase it is nite dimensional and if the input is a standard basis with respe t to the ring ordering. With two arguments: omputes the part of a ve tor spa e basis of the respe tive quotient with degree of the monomials equal to the se ond argument. Here, the quotient does not need to be nite dimensional. kill: deletes obje ts. killattrib: deletes the attribute given as the se ond argument koszul: koszul(d,n) omputes a matrix of the Koszul relations of degree d of the rst n ring variables. koszul(d,id) omputes a matrix of the Koszul relations of degree d of the generators of the ideal id. koszul(d,n,id) omputes a matrix of the Koszul relations of degree d of the rst n generators of the ideal id. laguerre: laguerre(p,n,m) omputes all omplex roots of the univariate polynomial p using Laguerre's algorithm. The se ond argument, n, de nes the pre ision of the fra tional part if the ground eld is the eld of rational numbers, otherwise it will be ignored. The third argument ( an be 0, 1 or 2) gives the number of extra runs for Laguerre's algorithm (with orrupted roots), leading to better results. lead: lead(I) returns the leading term(s) of a polynomial, a ve tor, respe tively of the generators of an ideal or module I with respe t to the monomial ordering. lead oef: lead oef(f) returns the leading oeÆ ient of a polynomial or a ve tor f with respe t to the monomial ordering.

B.5 Fun tions

leadexp:

519

leadexp(f) returns the exponent ve tor of the leading monomial of a polynomial or a ve tor f . In the ase of a ve tor the last omponent is the index in the ve tor. leadmonom: leadmonom(f) returns the leading monomial of a polynomial or a ve tor f as a polynomial or ve tor, whose oeÆ ient is one. LIB: reads a library of pro edures from a le. If the given lename does not start with . or / and annot be lo ated in the urrent dire tory, ea h dire tory ontained in the Sear hPath for libraries is sear hed for a le of this name. lift: lift(m,sm) omputes the transformation matrix, whi h expresses the generators of a submodule in terms of the generators of a module. More pre isely, if m denotes the module (or ideal), if sm denotes the submodule (or subideal), and if T denotes the transformation matrix returned by lift, then matrix(sm)*U=matrix(m)*T, where U is a diagonal matrix of units. U is always the unity matrix if the basering is a polynomial ring (not a power series ring). U is stored in the optional third argument. liftstd: liftstd(m,T) returns a standard basis of an ideal or module and the transformation matrix from the given ideal, respe tively module, to the standard basis. That is, if m is the ideal or module, sm the standard basis returned by liftstd, and T the transformation matrix, then matrix(sm)=matrix(m)*T. listvar: lists all (user{)de ned names in the urrent namespa e: listvar(): all urrently visible names ex ept pro edures, listvar(type): all urrently visible names of the given type, listvar(ring_name): all names, whi h belong to the given ring, listvar(name): the obje t with the given name, listvar(all): all names ex ept level of variables in pro edures is shown in square bra kets. listvar(pro ): all names of urrently available library pro edures. lres: omputes a free resolution of a homogeneous ideal using La S ala's algorithm. It an be used in the same way as hres. maxideal: maxideal(i) returns the i{th power of the maximal ideal generated by all ring variables (maxideal(i)=1 for i0 display poly/... tting to pagewidth [size n℄ print form-matrix [ rst n hars of ea h olumn℄ read Ma aulay_1 output and return its Singular format display any obje t in a ompa t format display id re ursively with respe t to variables in p split given string into lines of length n string of n spa e tabs write a list into a le and keep the list stru ture stop the omputation until user input

LIBRARY: intprog.lib Integer Programming with Grobner Basis Methods AUTHOR: Christine Theis, theismath.uni-sb.de PROCEDURES: solve_IP(..);

pro edures for solving integer programming problems

LIBRARY: latex.lib Typesetting of Singular-Obje ts in LaTeX2e AUTHOR: Christian Gorzel, gorzel math.uni-muenster.de PROCEDURES:

losetex(fnm); opentex(fnm); tex(fnm); texdemo([n℄); texfa torize(fnm,f); texmap(fnm,m,r1,r2); texname(fnm,s); texobj(l); texpoly(f,n[,l℄);

writes losing line for LaTeX-do ument writes header for LaTeX- le fnm

alls LaTeX2e for LaTeX- le fnm produ es a le explaining the features of this lib

reates string in LaTeX-format for fa tors of poly f

reates string in LaTeX-format for map m: r1 ! r2

reates string in LaTeX-format for identi er

reates string in LaTeX-format for any (basi ) type

reates string in LaTeX-format for poly

536


texpro (fnm,p); texring(fnm,r[,l℄); rmx(s); xdvi(s);

reates string in LaTeX-format of text from pro p

reates string in LaTeX-format for ring/qring removes .aux and .log les of LaTeX- les

alls xdvi for dvi- les

LIBRARY: linalg.lib Algorithmi Linear Algebra AUTHORS: Ivor Saynis h, ivsmath.tu- ottbus.de Mathias S hulze, ms hulzemathematik.uni-kl.de PROCEDURES: inverse(A); inverse_B(A); inverse_L(A); sym_gauss(A); orthogonalize(A); diag_test(A); busadj(A);

harpoly(A,v); adjoint(A); det_B(A); gaussred(A); gaussred_pivot(A); gauss_nf(A); mat_rk(A); U_D_O(A); pos_def(A,i); jordan(M[,opt℄); jordanmatrix(l); jordanform(M);

the inverse matrix of A list(matrix Inv,poly p), Inv A = p En (using busadj(A)) list(matrix Inv,poly p), Inv A = p En (using lift) symmetri Gaussian algorithm Gram-S hmidt orthogonalization test whether A an be diagonalized

oes of Adj(En t A) and oes of det(En t A)

hara teristi polynomial of A (using busadj(A)) adjoint of A (using busadj(A)) determinant of A (using busadj(A)) Gaussian redu tion: P A = U S, S row redu ed form of A Gaussian redu tion: P A = U S, uses row pivoting Gaussian normal form of A rank of onstant matrix A P A = U D O, P,D,U,O = permutation, diagonal, lower-, upper-triangular matrix test symmetri matrix for positive de niteness eigenvalues, Jordan blo k sizes, transformation matrix Jordan matrix with eigenvalues, Jordan blo k sizes Jordan normal form of onstant square matrix M

LIBRARY: LLL.lib Integral LLL-Algorithm (see also [122℄) AUTHOR: Alberto Vigneron-Tenorio, alberto.vigneronu a.es Alfredo San hez-Navarro, alfredo.san hezu a.es PROCEDURES: LLL(..);

Integral LLL-Algorithm

LIBRARY: matrix.lib

Elementary Matrix Operations

PROCEDURES:

ompress(A); matrix, zero olumns from A deleted

on at(A1,A2,..); matrix, on atenation of matri es A1,A2,... diag(p,n); matrix, nxn diagonal matrix with entries poly p dsum(A1,A2,..); matrix, dire t sum of matri es A1,A2,... flatten(A); ideal, generated by entries of matrix A generi mat(n,m[,id℄); generi nxm matrix [entries from id℄ is_ omplex( ); 1 if list is a omplex, 0 if not outer(A,B); matrix, outer produ t of matri es A and B power(A,n); matrix/intmat, n-th power of matrix/intmat A

B.8 Libraries skewmat(n[,id℄); submat(A,r, ); symmat(n[,id℄); tensor(A,B); unitmat(n); gauss_ ol(A); gauss_row(A); add ol(A, 1,p, 2); addrow(A,r1,p,r2); mult ol(A, ,p); multrow(A,r,p); perm ol(A,i,j); permrow(A,i,j); rowred(A[,any℄);

olred(A[,any℄); rm_unitrow(A); rm_unit ol(A);

537

generi skew-symmetri nxn matrix [entries from id℄ submatrix of A with rows/ ols spe i ed by intve r/ generi symmetri nxn matrix [entries from id℄ matrix, tensor produ t of matri es A and B unit square matrix of size n transform a matrix into ol-redu ed Gau normal form transform a matrix into row-redu ed Gau normal form add p ( 1-th ol) to 2-th olumn of matrix A, p poly add p (r1-th row) to r2-th row of matrix A, p poly multiply -th olumn of A with poly p multiply r-th row of A with poly p permute i-th and j-th olumns permute i-th and j-th rows redu tion of matrix A with elementary row-operations redu tion of matrix A with elementary ol-operations remove unit rows and asso iated olumns of A remove unit olumns and asso iated rows of A

LIBRARY: mondromy.lib Monodromy of an Isolated Hypersurfa e Singularity AUTHOR: Mathias S hulze, ms hulzemathematik.uni-kl.de OVERVIEW:

A library to ompute the monodromy of an isolated hypersurfa e singularity. It uses an algorithm by Brieskorn [24℄ to ompute a onne tion matrix of the meromorphi Gau-Manin onne tion up to arbitrarily high order, and an algorithm of Gerard and Levelt [69℄ to transform it to a simple pole. PROCEDURES: detadj(U); invunit(u,n); ja oblift(f); monodromyB(f[,opt℄); H2basis(f);

determinant and adjoint matrix of square matrix U series inverse of polynomial u up to order n lifts f in ja ob(f) with minimal monodromy of isolated hypersurfa e singularity f basis of Brieskorn latti e H''

LIBRARY: mprimde .lib Pro edures for Primary De omposition of Modules AUTHORS: Alexander Dreyer, adreyerweb.de REMARK:

These pro edures are implemented to be used in hara teristi 0. They also work in positive hara teristi 0. In small hara teristi and for algebrai extensions, the pro edures via Gianni, Trager, Za harias may not terminate. PROCEDURES: separator(l); Primde A(N[,i℄); Primde B(N,p); modDe (N[,i℄);

omputes a list of separators of prime ideals (not ne essarily minimal) primary de omposition via Shimoyama/Yokoyama (suggested by Grabe) (not ne essarily minimal) primary de omposition for pseudo-primary ideals minimal primary de omposition via Shimoyama/Yokoyama (suggested by Grabe)

538


minimal zero-dimensional primary de omposition via Gianni, Trager and Za harias GTZmod(N[, he k℄); minimal primary de omposition via Gianni, Trager and Za harias de 1var(N[, he k[,ann℄℄); primary de omposition for one variable annil(N); the annihilator of R^n/N in the basering splitting(N[, he k[,ann℄℄); splitting to simpler modules primTest(i[,p℄); tests whether i is prime or homogeneous preComp(N, he k[,ann℄); enhan ed version of splitting indSet(i); lists with varstrings of(in)dependent variables GTZopt(N[, he k[,ann℄℄); a faster version of GTZmod zeroOpt(N[, he k[,ann℄℄); a faster version of zeroMod

lrSBmod(N); extra ts an minimal SB from a SB minSatMod(N,I); minimal saturation of N w.r.t. I spe ialModulesEqual(N1,N2); he ks for equality of standard bases of modules if N1 is ontained in N2 or vi e versa stdModulesEqual(N1,N2);

he ks for equality of standard bases modulesEqual(N1,N2);

he ks for equality of modules getData(N,l[,i℄); extra ts oldData and omputes the remaining data zeroMod(N[, he k℄);

LIBRARY: mregular.lib Castelnuovo-Mumford Regularity of CM-S hemes AUTHORS: Isabel Bermejo, ibermejoull.es Philippe Gimenez, pgimenezagt.uva.es Gert-Martin Greuel, greuelmathematik.uni-kl.de OVERVIEW:

A library for omputing the Castelnuovo{Mumford regularity of a subs heme of the proje tive n-spa e that DOES NOT require the omputation of a minimal graded free resolution of the saturated ideal de ning the subs heme. The pro edures are based on [20℄, and [21℄. The algorithm assumes the variables to be in Noether position. PROCEDURES: reg_CM(id); regularity of arith. C-M subs heme V(id_sat) of Pn reg_ urve(id,[,e℄); regularity of proje tive urve V(id_sat) in Pn reg_mon urve(li); regularity of proje tive monomial urve de ned by li LIBRARY: normal.lib Normalization of AÆne Rings AUTHORS: Gert-Martin Greuel, greuelmathematik.uni-kl.de, Gerhard Pfister, pfistermathematik.uni-kl.de PROCEDURES: normal(I[,"wd"℄); HomJJ(L); genus(I);

omputes the normalization of basering/I, respe tively the normalization of basering/I and the delta-invariant presentation of End_R(J) as aÆne ring, L a list

omputes the genus of the proje tive urve de ned by I

LIBRARY: ntsolve.lib Real Newton Solving of Polynomial Systems AUTHORS: Wilfred Pohl, pohlmathematik.uni-kl.de Dietmar Hillebrand

B.8 Libraries

539

PROCEDURES: nt_solve(G,ini,[..℄); nd one real root of 0-dimensional ideal G triMNewton(G,a,[..℄); nd one real root for 0-dim triangular system G LIBRARY: paramet.lib Parametrization of Varieties AUTHOR: Thomas Keilen, keilenmathematik.uni-kl.de OVERVIEW:

A library to ompute parametrizations of algebrai varieties (if possible) with the aid of a normalization, or a primary de omposition, respe tively to ompute a parametrization of a plane urve singularity with the aid of a Hamburger-Noether expansion. PROCEDURES: parametrize(I); parametrizepd(I); parametrizesing(f);

parametrizes a prime ideal via the normalization

omputes primary de . and parametrizes omponents parametrizes an isolated plane urve singularity

LIBRARY: poly.lib Pro edures for Manipulating Polys, Ideals, Modules AUTHORS: Olaf Ba hmann, oba hmanmathematik.uni-kl.de, Gert-Martin Greuel, greuelmathematik.uni-kl.de, Anne Fruhbis-Kruger, annemathematik.uni-kl.de PROCEDURES: hilbPoly(I);

y li (int); katsura([i℄); freerank(poly/...) is_homog(poly/...); is_zero(poly/...); l m(ideal); max oef(poly/...); maxdeg(poly/...); maxdeg1(poly/...); mindeg(poly/...); mindeg1(poly/...); normalize(poly/...); rad_ on(p,I);

ontent(f); numerator(n); denominator(n) mod2id(M,iv); id2mod(i,iv); subrInterred(i1,i2,iv);

Hilbert polynomial of basering/I ideal of y li n-roots katsura [i℄ ideal rank of oker(input) if oker is free else -1 int, =1 resp. =0 if input is homogeneous resp. not int, =1 resp. =0 if oker(input) is 0 resp. not l m of given generators of ideal maximal length of oeÆ ient o

urring in poly/... int/intmat = degree/s of terms of maximal order int = [weighted℄ maximal degree of input int/intmat = degree/s of terms of minimal order int = [weighted℄ minimal degree of input normalize poly/... su h that leading oeÆ ient is 1

he k radi al ontainment of poly p in ideal I

ontent of polynomial/ve tor f numerator of number n denominator of number n

onversion of a module M to an ideal

onversion inverse to mod2id interred w.r.t. a subset of variables

LIBRARY: presolve.lib

Pre-Solving of Polynomial Equations

PROCEDURES: degreepart(id,d1,d2);

elements of id of total degree d1 and d2

540


elimlinearpart(id); elimpart(id[,n℄); elimpartanyr(i,p); fastelim(i,p[..℄); findvars(id[..℄); hilbve (id[, ,o℄); linearpart(id); tolessvars(id[,℄); solvelinearpart(id); sortandmap(id,s1,s2); sortvars(id[n1,p1..℄); shortid(id,n); valvars(id[..℄); idealSimplify(id); idealSplit(id,tF,fS);

linear part eliminated from id partial elimination of vars [among rst n vars℄ fa tors of p partially eliminated from i in any ring fast elimination of fa tors of p from i [options℄ ideal of variables o

urring in id [more information℄ intve of Hilbert series of id [in har and ord o℄ elements of id of total degree 1 maps id to new basering having only vars o

urring in id redu ed std-basis of linear part of id map to new basering with vars sorted w.r.t. omplexity sort vars w.r.t. omplexity in id [dierent blo ks℄ generators of id having at most n terms valuation of vars w.r.t. to their omplexity in id eliminates variables whi h are linear in id radi al of the interse tion of the ideals =radi al(id)

LIBRARY: primde .lib Primary De omposition and Radi al of Ideals AUTHORS: Gerhard Pfister, pfistermathematik.uni-kl.de (GTZ), Wolfram De ker, de kermath.uni-sb.de (SY), Hans S honemann, hannesmathematik.uni-kl.de (SY) OVERVIEW:

Algorithms for primary de omposition based on ideas of Gianni, Trager and Za harias [72℄ (implementation by G. P ster), respe tively based on ideas of Shimoyama and Yokoyama [167℄ (implementation by W. De ker and H. S honemann). The pro edures are implemented to be used in hara teristi 0. They also work in positive

hara teristi 0. In small hara teristi and for algebrai extensions, primde GTZ and minAssGTZ may not terminate, while primde SY and minAssChar may not give a omplete de omposition. Algorithms for the omputation of the radi al based on the ideas of Kri k, Logar [111℄ and Kemper (implementation by G. P ster). PROCEDURES: Ann(M); primde GTZ(I); primde SY(I...); minAssGTZ(I); minAssChar(I...); testPrimary(L,k); radi al(I); radi alEHV(I); equiRadi al(I); prepareAss(I); equidim(I); equidimMax(I); equidimMaxEHV(I); zerode (I);

annihilator of module Rn =M

omplete primary de omposition via Gianni,Trager,Za harias

omplete primary de omposition via Shimoyama-Yokoyama the minimal asso iated primes via Gianni,Trager,Za harias the minimal asso iated primes using hara teristi sets tests the result of the primary de omposition

omputes the radi al of I via Kri k/Logar and Kemper

omputes the radi al of I via Eisenbud,Huneke,Vas on elos the radi al of the equidimensional part of the ideal I list of radi als of the equidimensional omponents of I weak equidimensional de omposition of I equidimensional lo us of I equidimensional lo us of I via Eisenbud,Huneke,Vas on elos zero-dimensional de omposition via Moni o

B.8 Libraries

541

Computing a Primitive Element

LIBRARY: primitiv.lib AUTHOR: Martin Lamm,

lammmathematik.uni-kl.de

PROCEDURES: primitive(ideal i); primitive_extra(i); splitring(f,R[,L℄);

nd minimal polynomial for a primitive element nd primitive element for two generators de ne ring extension with name R and swit h to it

LIBRARY: qhmoduli.lib Moduli Spa es of Semi-Quasihomogeneous Singularities AUTHOR: Thomas Bayer, bayertin.tum.de OVERVIEW:

Compute equations for the moduli spa e of an isolated semi-quasihomogeneous hypersurfa e singularity with xed prin ipal part (based on [84℄). PROCEDURES: ArnoldA tion(f,[G,w℄); ModEqn(f); QuotientEquations(G,A,I); StabEqn(f); StabEqnId(I,w); StabOrder(f); UpperMonomials(f,[w℄); Max(data); Min(data); Table( md,i,lb,ub); LIBRARY: random.lib

indu ed a tion of G_f on T_ equations of the moduli spa e for prin ipal part f equations of Variety(I)/G w.r.t. a tion 'A' equations of the stabilizer of f equations of the stabilizer of the qhom. ideal I order of the stabilizer of f upper basis of the Milnor algebra of f maximal integer ontained in 'data' minimal integer ontained in 'data' list, i-th entry is md(i), lb i ub

Creating Random and Sparse Matri es, Ideals, Polys

PROCEDURES: generi id(i[,p,b℄); generi sparse linear ombinations of generators of i randomid(id,[k,b℄); random linear ombinations of generators of id randommat(n,m[,id,b℄); nxm matrix of random linear ombinations of id sparseid(k,u[,o,p,b℄); ideal of k random sparse poly's of degree d sparsematrix(n,m,o[,.℄); nxm sparse matrix of polynomials of degree o sparsemat(n,m[,p,b℄); nxm sparse integer matrix with random oeÆ ients sparsepoly(u[,o,p,b℄); random sparse polynomial, terms of degree in [u,o℄ sparsetriag(n,m[,.℄); nxm sparse lower-triang intmat with random oes triagmatrix(n,m,o[,.℄); nxm sparse lower-triang matrix of poly's of deg o randomLast(b); random transformation of the last variable randomBinomial(k,u,..); binomial ideal, k random generators of deg u LIBRARY: rees los.lib AUTHOR: Tobias Hirs h,

Pro edures to Compute Integral Closure of an Ideal

hirs hmath.tu- ottbus.de

OVERVIEW:

A library to ompute the integral losure of an ideal I in a R=k[x(1),...,x(n)℄ using the Rees{Algebra R[It℄ of I. It omputes the integral losure of R[It℄ (in the same

542


manner as done in the library 'normal.lib'), whi h is a graded subalgebra of R[t℄. The degree-k- omponent is the integral losure of the k-th power of I. PROCEDURES: ReesAlgebra(I); normalI(I[,p[,r℄℄);

omputes Rees-Algebra of an ideal I

omputes integral losure of an ideal I using R[It℄

LIBRARY: ring.lib

Manipulating Rings and Maps

PROCEDURES:

hange har("R", [,r℄);

hangeord("R",o[,r℄);

hangevar("R",v[,r℄); defring("R", ,n,v,o); defrings(n[,p℄); defringp(n[,p℄); extendring("R",n,v,o); fet hall(R[,str℄); imapall(R[,str℄); mapall(R,i[,str℄); ord_test(R); ringtensor("R",s,t,..); ringweights(r);

make a opy R of basering [ring r℄ with new har make a opy R of basering [ring r℄ with new ord o make a opy R of basering [ring r℄ with new vars v de ne a ring R in spe i ed har , n vars v, ord o de ne ring Sn in n vars, har 32003 [p℄, ord ds de ne ring Pn in n vars, har 32003 [p℄, ord dp extend given ring by n vars v, ord o and name it R fet h all obje ts of ring R to basering imap all obje ts of ring R to basering map all obje ts of ring R via ideal i to basering test whether ordering of R is global, lo al or mixed

reate ring R, tensor produ t of rings s,t,... intve of weights of ring variables of ring r

LIBRARY: rinvar.lib AUTHOR: Thomas Bayer,

tbayerin.tum.de

Invariant Rings of Redu tive Groups

OVERVIEW:

Implementation based on Derksen's algorithm. Written in the frame of the diploma thesis (advisor: Prof. Gert-Martin Greuel) 'Computations of moduli spa es of semiquasihomogeneous singularities and an implementation in Singular' PROCEDURES: HilbertSeries(I, w); HilbertWeights(I, w); ImageVariety(I, F); ImageGroup(G, F); InvariantRing(G, Ga tion); InvariantQ(f, G, Ga tion); LinearizeA tion(G, Ga tion); LinearA tionQ(a tion,s,t); LinearCombinationQ(base, f); MinimalDe omposition(f,s,t); NullCone(G,a t); ReynoldsImage(RO,f); ReynoldsOperator(G, Ga tion); SimplifyIdeal(I[,m,s℄); TransferIdeal(R,name,nA);

Hilbert series of the ideal I w.r.t. weight w weighted degrees of the generators of I ideal of the image variety F(variety(I)) ideal of G w.r.t. the indu ed representation generators of the invariant ring of G de ide if f is invariant w.r.t. G linearization of the a tion 'Ga tion' of G de ide if a tion is linear in var(s..nvars) de ide if f is in the linear hull of 'base' minimal de omposition of f (like oef) ideal of the null one of the a tion 'a t' of G image of f under the Reynolds operator 'RO' Reynolds operator of the group G simplify the ideal I (try to redu e variables) transfer the ideal 'name' from R to basering

B.8 Libraries

543

LIBRARY: sing.lib Invariants of Singularities AUTHORS: Gert-Martin Greuel, greuelmathematik.uni-kl.de, Bernd Martin, martinmath.tu- ottbus.de PROCEDURES:

odim (id1, id2); deform(i); dim_slo us(i); is_a tive(f,id); is_ i(i); is_is(i); is_reg(f,id); is_regs(i[,id℄); milnor(i); nf_i is(i); qhspe trum(f,w); slo us(i); tangent one(i); Tjurina(i); tjurina(i); T_1(i); T_2((i); T_12(i);

ve tor spa e dimension of of id2/id1 if nite in nitesimal deformations of ideal i dimension of singular lo us of ideal i is poly f an a tive element mod id? (id ideal/module) is ideal i a omplete interse tion? is ideal i an isolated singularity? is poly f a regular element mod id? (id ideal/module) are gen's of ideal i regular sequen e modulo id? Milnor number of ideal i; (assume i is ICIS in nf) generi ombinations of generators; get ICIS in nf spe trum numbers of w-homogeneous polynomial f ideal of singular lo us of ideal i tangent one of ideal i SB of Tjurina module of ideal i (assume i is ICIS) Tjurina number of ideal i (assume i is ICIS) T^1-module of ideal i T^2-module of ideal i T^1- and T^2-module of ideal i

LIBRARY: solve.lib Complex Solving of Polynomial Systems AUTHOR: Moritz Wenk, wenkmathematik.uni-kl.de Wilfred Pohl, pohlmathematik.uni-kl.de PROCEDURES: laguerre_solve(p,[..℄); solve(i,[..℄); ures_solve(i,[..℄); mp_res_mat(i,[..℄); interpolate(p,v,d); fglm_solve(i,[..℄); lex_solve(i,p,[..℄); triangLf_solve(l,[..℄); triangM_solve(l,[..℄); triangL_solve(l,[..℄); triang_solve(l,p,[..℄);

nd all roots of univariate polynomial p all roots of 0-dim. ideal i using triangular sets nd all roots of 0-dimensional ideal i with resultants multipolynomial resultant matrix of ideal i interpolate poly from evaluation points p and results v nd roots of 0-dim. ideal using FGLM and lex_solve nd roots of redu ed lexi ographi standard basis nd roots using triangular sys. (fa torizing Lazard) nd roots of given triangular system (Moeller) nd roots using triangular system (Lazard) nd roots of given triangular system

LIBRARY: sp urve.lib Deformations and Invariants of CM- odim 2 Singularities AUTHOR: Anne Fruhbis-Kruger, annemathematik.uni-kl.de PROCEDURES: isCM od2(i); CMtype(i); matrixT1(M,n); semiCM od2(M,T1);

presentation matrix of the ideal i, if i is CM Cohen-Ma aulay type of the ideal i 1-st order deformation T1 in matrix des ription semiuniversal deformation of maximal minors of M

544


dis r(sem,n); qhmatrix(M); relweight(N,W,a); posweight(M,T1,i); KSpen erKernel(M);

dis riminant of semiuniversal deformation weights if M is quasihomogeneous relative matrix weight of N w.r.t. weights (W,a) deformation of oker(M) of non-negative weight kernel of the Kodaira-Spen er map

LIBRARY: spe trum.lib AUTHOR: Stefan Endra

Singularity Spe trum for Nondegenerate Singularities

PROCEDURES: spe trumnd(poly[,1℄); semi ont(s1,s2[,1℄); semi ontqh(s1,s2); spadd(s1,s2); spmul(s,k);

spe trum of a nondegenerate isolated singularity tests if s2 is semi ontinuous for s1 semi ontinuity test using open and half open intervals sum of two spe tra s1 and s2 produ t of spe trum s with integer k

LIBRARY: standard.lib

Pro edures, whi h are always loaded at Start-up

PROCEDURES: stdfglm(ideal[,ord℄) stdhilb(ideal[,h℄) groebner(ideal/module) quot(any,any[,n℄) res(ideal/module,[i℄) sprintf(fmt,...) fprintf(link,fmt,..) printf(fmt,...)

standard basis of ideal via fglm [and ordering ord℄ standard basis of ideal using the Hilbert fun tion standard basis using a heuristi ally hosen method quotient using heuristi ally hosen method free resolution of ideal or module returns formatted string writes formatted string to link displays formatted string

LIBRARY: stratify.lib Algorithmi Strati ation for Unipotent Group-A tions AUTHOR: Anne Fruhbis-Kruger, annemathematik.uni-kl.de OVERVIEW:

This library provides an implementation of the algorithm of Greuel and P ster introdu ed in the arti le \Geometri quotients of unipotent group a tions". PROCEDURES: prepMat(M,wr,ws,step); list of submatri es orresp. to given ltration stratify(M,wr,ws,step); algorithmi strati ation (main pro edure) LIBRARY: surf.lib Pro edures for Graphi s with surf AUTHOR: Hans S honemann, hannesmathematik.uni-kl.de, the program surf is written by Stefan Endra NOTE:

To use this library requires the program surf to be installed. surf is only available for Linux PCs and Sun workstations. You an download surf either from http://sour eforge.net/proje ts/surf

or from ftp://www.mathematik.uni-kl.de/pub/Math/Singular/utils/.

B.8 Libraries PROCEDURES: plot(I,[...℄);

545

plots plane urves and surfa es

LIBRARY: tea hstd.lib Pro edures for Tea hing Standard Bases AUTHOR: Gert-Martin Greuel, greuelmathematik.uni-kl.de NOTE:

The library is intended to be used for tea hing purposes only. The pro edures are implemented exa tly as des ribed in the book 'A SINGULAR Introdu tion to Commutative Algebra' by G.-M. Greuel and G. P ster. SuÆ iently high printlevel allows to ontrol ea h step. PROCEDURES: e art(f); tail(f); sameComponent(f,g); leadmonomial(f); monomialL m(m,n); spoly(f[,1℄); minE art(T,h); NFMora(i); prod rit(f,g);

hain rit(f,g,h); pairset(G); updatePairs(P,S,h); standard(id); lo alstd(id);

e art of f tail of f test for same module omponent of lead(f) and lead(g) leading monomial as poly (also for ve tors) l m of monomials m and n as poly (also for ve tors) s-polynomial of f [symmetri form℄ element g 2 T of minimal e art s.t. LM(g) divides LM(h) normal form of i w.r.t Mora algorithm test for produ t riterion test for hain riterion pairs form G neither satisfying prod rit nor hain rit pairset P enlarged by not useless pairs (h,f), f in S standard basis of ideal/module lo al standard basis of id using Lazard's method

LIBRARY: template.lib A Template for a Singular Library AUTHOR: Olaf Ba hmann, oba hmanmathematik.uni-kl.de PROCEDURES: mdouble(int); mtripple(int); msum([int,..,int℄);

returns double of int argument returns three times int argument sum of int arguments

LIBRARY: tori .lib Standard Basis of Tori Ideals AUTHOR: Christine Theis, theismath.uni-sb.de PROCEDURES: tori _ideal(A,..); omputes the tori ideal of A tori _std(ideal I); standard basis of I by a spe ialized Bu hberger algorithm LIBRARY: triang.lib De ompose Zero-dimensional Ideals into Triangular Sets AUTHOR: Dietmar Hillebrand PROCEDURES: triangL(G); triangLfak(G); triangM(G[,.℄);

De omposition of (G) into triangular systems (Lazard). De omp. of (G) into tri. systems plus fa torization. De omposition of (G) into triangular systems (Moller).

546


triangMH(G[,.℄);

De omp. of (G) into tri. syst. with disjoint varieties.

LIBRARY: weierstr.lib Pro edures for the Weierstra Theorems AUTHOR: Gert-Martin Greuel, greuelmathematik.uni-kl.de PROCEDURES: weierstr_div(g,f,d); weierstr_prep(f,d); lastvar_general(f); general_order(f);

perform Weierstrass division of g by f up to gegree d perform Weierstrass preparation of f up to gegree d make f general of nite order w.r.t. last variable

ompute integer b s.t. f is x_n-general of order b

LIBRARY: zeroset.lib Pro edures For Roots and Fa torization AUTHOR: Thomas Bayer, bayertin.tum.de OVERVIEW:

Algorithms for nding the zero{set of a zero{dimensional ideal in Q(a)[x1 ; ::; xn ℄, Roots and Fa torization of univariate polynomials over Q(a)[t℄ where a is an algebrai number. Written in the frame of the diploma thesis (advisor: Prof. GertMartin Greuel) 'Computations of moduli spa es of semiquasihomogeneous singularities and an implementation in Singular'. This library is meant as a preliminary extension of the fun tionality of Singular for univariate fa torization of polynomials over simple algebrai extensions in hara teristi 0. Subpro edures with post x 'Main' require that the ring ontains a variable 'a' and no parameters, and the ideal 'mpoly', where 'minpoly' from the basering is stored. PROCEDURES: EGCD(f,g); Fa tor(f); Quotient(f,g); Remainder(f,g); Roots(f); SQFRNorm(f); ZeroSet(I);

g d over an algebrai extension eld of Q fa torization of f over an algebrai extension eld quotient q of f w.r.t. g (in f = q g + remainder) remainder of the division of f by g

omputes all roots of f in an extension eld of Q norm of f (f must be squarefree) zero-set of the 0-dim. ideal I

AUXILIARY PROCEDURES: EGCDMain(f,g); Fa torMain(f); InvertNumberMain( ); QuotientMain(f,g); RemainderMain(f,g); RootsMain(f); SQFRNormMain(f); ContainedQ(data,f); SameQ(a,b);

g d over an algebrai extension eld of Q fa torization of f over an algebrai extension eld inverts an element of an algebrai extension eld quotient of f w.r.t. g remainder of the division of f by g

omputes all roots of f, might extend the ground eld norm of f (f must be squarefree) f in data ? a = b (list a,b)?

B.9 Singular and Maple In this se tion we give two examples how to use Singular as a support for a Maple session and one example showing how to use Maple in a Singular

B.9

Singular and Maple

547

session. The rst example is the trivial way, the se ond is based on a simpli ed version of a s ript of G. Kemper.6 Assume we are in a Maple session and want to ompute a Grobner basis with Singular of the ideal I = hx10 + x9 y 2 ; y 8 x2 y 7 i in hara teristi 0 with the degree reverse lexi ographi al ordering dp. The rst solution is to write the polynomials to the le singular input (already in the Singular language). This is done by the following: f:=x^10+x^9*y^2; g:=y^8-x^2*y^7; interfa e(prettyprint=0); interfa e(e ho=0); writeto( singular_input ); lprint(ìdeal I = `); f, g ; lprint(`;`); writeto(terminal);

The resulting le looks like: ideal I = x^10+x^9*y^2, y^8-x^2*y^7 ;

Now we an start Singular, and perform the following ring R=0,(x,y),dp; < "singular_input"; short=0; // output in Maple format ideal J=std(I); write(":w maple_input",J);

This Singular session writes the omputed Grobner basis (in Maple format) to the le maple input: x^2*y^7-y^8,x^9*y^2+x^10,x^12*y+x*y^11,x^13-x*y^12,y^14+x*y^12, x*y^13+y^12

A more advan ed solution is given by the following pro edure. Here I is a list of polynomials, P is the hara teristi of the ground eld, and tord is a string spe ifying the Singular ordering (for instan e dp, lp, : : : ).7 6 Warning: The s ripts run only on Unix like operating systems. 7 This pro edure works with Maple V Release 5. In older versions of Maple, string expression were en losed in a pair of ba k quotes ` ` instead of " "; moreover, the nullary operator was denoted by " instead of %. The dire tory EXAMPLES/ on the en losed CD ontains two versions of the pro edure { one for Maple V Release 5 and one for Maple V Release 3 (with the old syntax).

548

B. SINGULAR | A Short Introdu tion SINGULARlink:=pro (I,P,tord) lo al i,j,path,vars,F,p,ele; F:=map(expand,I); vars:=indets(F); if P>0 then p:=P else p:=0 fi; path:="Trans"; if assigned(pathname) then path:= at(pathname,path) fi; if system("mkdir ".path)0 then ERROR("Couldn't make the Transfer-dire tory") fi; # produ e input for Singular ... writeto( at(path,"/In")); # Define the ring (with term order) lprint(`ring R =`); lprint(``.p.`,`); lprint(`(x(1..`.(nops(vars)).`)),`); lprint(`(`.tord.`);`); # Define the ideal ... lprint(ìdeal I =`); for i to nops(F) do if i>1 then lprint(`,`) fi; ele:=subs([seq(vars[j℄=x(j), j=1..nops(vars))℄,F[i℄); if type(ele,monomial) then lprint(ele) else # split into summands, otherwise line might get hopped! lprint(op(1,ele)); for j from 2 to nops(ele) do lprint(`+`,op(j,ele)) od fi od; lprint(`;`); lprint(`short=0;`); lprint(ìdeal b = std(I);`); lprint(`write(\"`.path.`/Out\",\"[\");`); lprint(`write(\"`.path.`/Out\",b);`); lprint(`write(\"`.path.`/Out\",\"℄;\");`); lprint(`quit;`); writeto(terminal); # Call Singular ... system("Singular < ".path."/In > ".path."/temp"); if %0 then system("'rm' -r ".path); ERROR("Something went wrong while exe uting Singular") fi; # Retrieve the results ...

B.9

Singular and Maple

549

read( at(path,"/Out")); F:=%; system("'rm' -r ".path); F:=subs([seq(x(i)=vars[i℄,i=1..nops(vars))℄,F); F:=map(expand,F); F end:

Let's apply this pro edure to an example: f:=x^10+x^9*y^2; g:=y^8-x^2*y^7; J:=SINGULARlink([f,g℄,0,"dp"): interfa e(prettyprint=0); J; #-> [x^2*y^7-y^8, x^10+x^9*y^2, x^12*y+x*y^11, x^13-x*y^12, #-> y^14+x*y^12, x*y^13 +y^12℄

Assume now that we are in a Singular session and want to use Maple to fa torize a polynomial. This an be done by using the following Singular pro edure: pro maple_fa torize(poly p) { int saveshort=short; short=0; string in="maple-in."+string(system("pid")); string out="maple-out."+string(system("pid")); link l=in; write(l,"res:=fa tors("+string(p)+");"); write(l,"interfa e(prettyprint=0);"); write(l,"interfa e(e ho=0);"); write(l,"writeto(`"+out+"`);"); write(l,"lprint(ìdeal fa =`);"); write(l,"lprint(op(1,res));"); write(l,"for i to nops(op(2,res)) do"); write(l," lprint(`,`);"); write(l," lprint(op(1,op(i,op(2,res))));"); write(l,"od;"); write(l,"lprint(`;`);"); write(l,"lprint(ìntve multies=`);"); write(l,"lprint(`1`);"); write(l,"for i to nops(op(2,res)) do"); write(l," lprint(`,`);"); write(l," lprint(op(2,op(i,op(2,res))));"); write(l,"od;"); write(l,"lprint(`;`);"); write(l,"lprint(`list res=fa ,multies;`);"); write(l,"writeto(terminal);"); write(l,"quit;"); int dummy=system("sh","maple dummy"); if (dummy 0) { ERROR("something went wrong"); } string r=read(out);

550


}

exe ute(r); return(res);

Here omes an example: ring R = 0,(x,y),dp; poly f = 5*(x-y)^2*(x+y); maple_fa torize(f); //-> [1℄: //-> _[1℄=5 //-> _[2℄=x+y //-> _[3℄=-x+y //-> [2℄: //-> 1,1,2

B.10 Singular and Mathemati a We show by an example how to use Singular as support for a Mathemati a session.8 Assume we are in a Mathemati a session and want to ompute a Grobner basis with Singular of the ideal hx10 + x9 y 2 ; y 8 x2 y 7 i in hara teristi 0 with the degree reverse lexi ographi al ordering dp. SINGULARlink[J_List,P_Integer,tord_String℄ := Module[ {i,vars,F,p,subst,varnames,SINGULARin,SINGULARout}, F=J; vars=Variables[F℄; (* Substitution of Variable names *) varnames = Table[ ToExpression[ "x[" ToString[i℄ "℄"℄, {i, 1, Length[vars℄} ℄; subst = Dispat h[MapThread[Rule, {vars, varnames} ℄℄; F = F /. subst; singF = ToString[F // InputForm℄; singF = StringRepla e[ singF, { "{"->"", "}"->"", ", "->",\n", "℄"->")", "["->"(" } ℄; If[P>0, p=P, (*Else*) p=0℄; (* Prepare Singular input string *) SINGULARin = "ring R =" ToString[p℄ ", (" "x(1.." ToString[Length[vars℄℄ ")), (" tord ");\n"; (* Define the ideal... *) SINGULARin = SINGULARin "ideal I =" singF " ;\n" "short=0;\n" "ideal b = std(I);\n" "write(\".tmp.sing.mathemati a\",b );\n" "quit;\n"; (* Call Singular ... *) SINGULARin // OutputForm >> "!Singular -q";

8 Warning: The s ripts run only on Unix like operating systems.

B.11

℄

Singular and MuPAD

551

SINGULARout=ReadList[".tmp.sing.mathemati a",String℄; >>"!rm .tmp.sing.mathemati a"; SINGULARout= StringRepla e[SINGULARout , { ")"->"℄", "("->"[" } ℄; SINGULARout= "{" SINGULARout "}"; SINGULARout= ToExpression[SINGULARout℄; subst = Dispat h[MapThread[Rule, {varnames, vars} ℄℄; SINGULARout=SINGULARout/. subst; SINGULARout

Let's apply this f=x^10+x^9*y^2; g=y^8-x^2*y^7; J=SINGULARlink[{f,g},0,"dp"℄; J // InputForm (*-> //InputForm= {x^2*y^7 - y^8, x^10 + x^9*y^2, x^12*y + x*y^11, x^13 - x*y^12, x*y^12 + y^14, y^12 + x*y^13} *)

B.11 Singular and MuPAD Finally, we give an example how to use Singular as support for a MuPAD session.9 Assume we are in a MuPAD session and want to ompute a Grobner basis with Singular of the ideal hx10 + x9 y 2 ; y 8 x2 y 7 i in hara teristi 0 with the degree reverse lexi ographi al ordering dp. SINGULARlink := pro (L:DOM_LIST, P:DOM_INT, tord) lo al i, path, ipath, opath, tpath, fd, vars, F, p; begin F := map(L, expand); vars := indets(F); if P > 0 then p := P; else p := 0; end_if; // Create the dire tory where " ommuni ation files" are stored path := "Trans"; if singpath hold(singpath) then path := pathname(singpath, path) ;

9 We should like to thank Torsten Metzner for providing the s ripts. Warning: The s ripts run only on Unix like operating systems.

552

B. SINGULAR | A Short Introdu tion end_if; if system("mkdir ".path) 0 then error("Could not make the Transfer-dire tory") end_if; // produ e input for Singular ... ipath := pathname(path)."In" ; if version() = [2, 0, 0℄ then fd := fopen(Text, ipath, Write) ; else // MuPAD 2.5 fd := fopen(ipath, Write, Text) ; end_if ; // Define the ring (with term order) fprint(NoNL, fd, "ring R = "); fprint(NoNL, fd, p,", ("); fprint(NoNL, fd, vars[i℄, ", " ) $i=1..nops(vars)-1; fprint(NoNL, fd, vars[nops(vars)℄, "), "); fprint(Unquoted, fd, "( ",tord," ); "); // Define the ideal ... fprint(Unquoted, fd, "ideal I ="); fprint(Unquoted, fd, F[i℄, ", " ) $i=1..nops(F)-1; fprint(Unquoted, fd, F[nops(F)℄, ";"); fprint(Unquoted, fd, "short=0;"); fprint(Unquoted, fd, "ideal b = std(I);"); opath := pathname(path)."Out" ; fprint(Unquoted, fd, "write(\"".opath."\",\"[\");"); fprint(Unquoted, fd, "write(\"".opath."\",b);"); fprint(Unquoted, fd, "write(\"".opath."\",\"℄;\");"); fprint(Unquoted, fd, "quit;"); f lose(fd); // Call Singular ... tpath := pathname(path)."temp" ; if system("Singular < ".ipath."> ".tpath) 0 then system("rm -r ".path); error("Something went wrong while exe uting Singular"); end_if; // Retrieve the results ... F := read(opath, Quiet): system("rm -r ".path); F := map(F,expand); end_pro :

Now apply this pro edure: f:=x^10+x^9*y^2; g:=y^8-x^2*y^7; J:=SINGULARlink([f,g℄,0,"dp"): output::tableForm(J,",");

B.11

Singular and MuPAD

553

//-> x^2*y^7 - y^8,x^10 + x^9*y^2,x*y^11 + x^12*y,x^13 - x*y^12, //-> y^14 + x*y^12,y^12 + x*y^13

Assume now that we are in a Singular session and want to use MuPAD to fa torize a polynomial. This an be done by using the following Singuar pro edure: pro mupad_fa torize(poly p) { int saveshort = short; short = 0; string in = "mupad-in."+string(system("pid")); string out = "mupad-out."+string(system("pid")); link l = in; write(l, "res:=fa tor("+string(p)+");"); write(l, "if version() = [2, 0, 0℄ then "); write(l, "fd := fopen(Text, \""+out+"\", Write) ;"); write(l, "else"); write(l, "fd := fopen(\""+out+"\", Write, Text) ;"); write(l, "end_if;"); write(l,"fprint(Unquoted, fd, \"ideal fa =\");"); write(l,"fprint(Unquoted, fd, op(res,1));"); write(l,"for i from 2 to nops(res) step 2 do"); write(l," fprint(Unquoted, fd, \",\", op(res,i));"); write(l,"end_for;"); write(l,"fprint(Unquoted, fd, \";\");"); write(l,"fprint(Unquoted, fd, \"intve multies=\");"); write(l,"fprint(Unquoted, fd, \"1\");"); write(l,"for i from 3 to nops(res) step 2 do"); write(l," fprint(Unquoted, fd, \",\", op(res,i));"); write(l,"end_for;"); write(l,"fprint(Unquoted, fd, \";\");");

}

write(l,"fprint(Unquoted, fd, \"list res=fa ,multies;\");"); write(l,"f lose(fd);"); write(l,"quit;"); int dummy = system("sh","mupad dummy"); if (dummy 0) { ERROR("something went wrong"); } string r=read(out); exe ute(r); return(res);

We apply this pro edure to an example: ring R = 0,(x,y),dp; poly f = 5*(x-y)^2*(x+y); mupad_fa torize(f); //-> [1℄: //-> _[1℄=5

554

B. SINGULAR | A Short Introdu tion //-> _[2℄=x+y //-> _[3℄=-x+y //->[2℄: //-> 1,1,2

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Computer Algebra Systems 189. ASIR (Noro, M.; Shimoyama, T.; Takeshima, T.): http://www.asir.org/. 190. CALI (Grabe, H.-G.): A REDUCE Pa kage for Commutative Algebra. Institut fur Informatik, Universitat Leipzig. 191. CASA (Winkler, F. et al.): http://www.ris .uni-linz.a .at/software/ asa/

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194. Ma aulay (Bayer, D.; Stillman, M.): A System for Computation in Algebrai Geometry and Commutative Algebra. Available via ftp from zariski.harvard.edu . 195. Ma aulay 2 (Grayson, D.; Stillman, M.): A Computer Software System Designed to Support Resear h in Commutative Algebra and Algebrai Geometry. Available from http://math.uiu .edu/Ma aulay2 . 196. Ma syma (Ma syma In .): http://www.s ientek. om/ma syma/main.htm 197. Magma (Computational Algebra Group within S hool of Maths and Statisti s of University of Sydney): http://magma.maths.usyd.edu.au/magma

198. Maple (Waterloo Maple In .): http://www.maplesoft. om/.

564

Referen es 199. MAS (Kredel, H. ; Pes h, M.):

http://krum.rz.uni-mannheim.de/mas.html

200. Mathemati a (Wolfram Resear h In .): http://www.wolfram. om/. 201. MuPAD (MuPAD Resear h Group and S ifa e Software GmbH & Co. KG): http:/www.mupad.de/. 202. Redu e: http://www.uni-koeln.de/REDUCE. 203. Singular (Greuel, G.-M.; P ster, G.; S honemann, H.): A Computer Algebra System for Polynomial Computations. Centre for Computer Algebra, University of Kaiserslautern (2001). http://www.singular.uni-kl.de.

Glossary Glossary

(>; ), 118 (A; m), 31 (A; m; K ), 31 ( ; >), 118 (x0 : : : : : xn ), 433 (>1 ; >2 ), 14 >(A;>) , 12 >Dp , 13 >Ds , 14 >dp , 13 >ds , 14 >lp , 13 >ls , 14 >w , 18 A(d), 115 A[tI ℄, 232 A+ , 443 AP , 33 Af , 33 Ared , 27, 210 I a , 441 I h , 63, 64, 245, 441 I1 : I21 , 81 I1 I2 , 19 I1 , 441 K (x1 ; : : : ; xn ), 33 K [[x℄℄, 315 K [x℄+ , 435 K [x℄> , 39 K [x℄d , 434 K hx1 ; : : : ; xn i, 485 M (d), 115 M=N , 100 M , 93 MP , 111 MS , 111 Mf , 111 N : P , 101 N :A P , 101 P :M I , 101 S 1 M , 111

S>, 39 V , 441 V1 , 441 X Y , 464 X Y , 421 X1 \ X2 , 427 X1 [ X2 , 427 [a℄, 22

99 K

f , 421, 459

AjK , 314 Ln;m , 457 i2I Mi , 101

m, 31 ma, 31 x y, 14 hai, 1 hw; i, 13

(f ), 487 Q(f; p), 487 pi2I Mi , 101 M p N , 183, 192 I , 27 (f ), 487 (f; p), 487 '# , 418, 425 'S , 111 Ab, 327

, 327 M t A, 153 a + I , 22 f , 417, 427 f a, 438 f h, 56, 63, 64, 438 x ei j x ej , 119

A n , 412 A nK , 412

AHI (n), 291 Ann(P ), 101 AnnM (a), 350 AnnR (I ), 27

565

566

Glossary

AnnA (P ), 101 Ass(I ), 241 Ass(I; P ), 241 Ass(M ), 246

G(t), 278

B (M; N ; P ), 167 bj;k (M ), 137 bk (M ), 133, 137

HM , 277 H1 , 438, 441 Hi , 438 HC(I ), 60

C (I; B ), 201 CX , 435 Cp (X ), 479 C (A), 207 C (A; P ), 207 Cp X , 436 Coker('), 100 Crit(f ), 487 Cs (I; B ), 201 D(f ), 444, 450 D(xi ), 438 d(I; K [x℄), 221 d(M ), 285 deg(M ), 285 deg(X ), 475 deg(f ), 2 depth(I; M ), 372 depth(M ), 372 dim(A), 207 dim(I ), 207 dim(M ), 302 dim(X ), 416 dimp X , 471 En , 156, 160, 194 E (I ), 260 E (I ), 262 e art(f ), 56 e artw (f ), 56 edim(A), 301 ExtiA (M; N ), 261, 342 Fk (M ), 343 FkA (M ), 343 Flat(M ), 354 Flatr (M ), 351

G , 46

gldim(A), 400 GrI (A), 114

HN = Hilbert's Nullstellensatz, 415 HomA (M; N ), 92 HPM , 277 HSfMn gn0 , 294 HSPfMn gn0 , 295 ht(I ), 207

I (X ), 415, 419, 424, 435 i(C; D; p), 482 i(F; G; p), 482 i(f; g; p), 482 Im('), 99 In(I ), 298, 300 In(f ), 298, 300 IX (Y ), 419 jk (f ), 316 K (X ), 452 K (x) , 373 K (x; M ) , 373 K [X ℄, 416, 435 K [X ℄d, 435 K [X ℄(mp) , 473 Ker('), 99 L(G), 41, 119 L(I ), 41 L>(G), 41, 119 LC(f ), 10, 41, 118 l m(x ; x ), 49 LE(f ), 10, 41 lim !1 f , 317 LM(f ), 10, 41, 118 LT(f ), 10, 41, 118 Mat(m n; A), 93

Glossary

Max(A), 23, 423 minAss(I ), 208 Mon(x1 ; : : : ; xn ), 2 Monn , 2 Mor(X; Y ), 417 mult(M ), 296 mult(M; Q), 296 mult(X; p), 475 NF( j I ), 48 NF(f j G), 46, 120

O(X ), 448 an , 484 OX;p

ord(f ), 297, 315

PM , 279, 285 P(V ), 433

Pn, 433 PnK , 433 PnR , 444 PK (V ), 433

pdA (M ), 390, 393 Proj(A), 443

Q(A), 32, 33 Q(t), 278 QM (t), 280 Quot(A), 32, 33 R(X ), 451 rad(I ), 27 radM (N ), 183 sat(I ), 445 Sing(A), 308, 358 Sing(f ), 487 Sing(X ), 474 SolS (AZ = b), 188 Spe (A), 23, 423 spoly(f; g ), 49, 122 supp(I ), 82 supp(M ), 112 syz(I ), 140 syz(f1 ; : : : ; fk ), 140 syzR k (I ), 140

syzk (I ), 140 tail(f ), 10, 119 TorA i , 337 Tors(M ), 101 trdegK L, 218

V (I ), 112, 413, 423, 443 V (f ), 444 V (f )2 , 412 w{deg(f ), 13 w{deg(x ei ), 370 w{ord(f ), 315

567

568

Glossary

Index

569

Index

A1 {singularity, 489

A'Campo, 407 addition, 91 additive fun tion, 130 ADE{singularity, 492 aÆne, 449 algebrai set, 412 dimension of, 416, 452

hart, 438

one, 435, 444, 446 Hilbert fun tion, 291 part, 440, 441 ring, 23, 447 s heme, 427 spa e n{dimensional, 412 surfa e, 413 variety, 412 aÆnization, 438, 441 AG{ odes, 530 ainvar.lib, 530 algDependent, 87, 530 algebra, 3, 92 map, 92 algebra.lib, 20, 87, 198, 393, 420, 530 algebra ontainment, 87, 530 algebrai dependen e, 86 lo al ring, 484, 486 set, 404 aÆne, 412, 415 irredu ible, 416, 434 proje tive, 434 variety, 404 algebrai ally dependent, 86 alg kernel, 20, 530 A{linear, 92 analyti fun tion, 484 germ, 484 K {algebra, 323

lo al ring, 484, 486

Ann, 302, 395, 540

annihilator, 27, 101, 183, 185, 538, 540 stru ture, 430 arithmeti geometry, 423 stru ture, 446 Arnold, 492, 531 Artin{Rees, 293 Artinian K {algebra, 30 ring, 30 as ending hain ondition, 22 asso iated prime, 241, 409 embedded, 241 minimal, 208 of a module, 246 of a ring, 241 attrib, 117, 138, 239, 512 attribute, 117, 138, 239 Auslander, 400 Auslander{Bu hsbaum formula, 390, 394, 397 automorphism, 20 bareiss, 512

Barth sexti , 405 basis, 102

anoni al, 93 Di kson, 12 betti, 150, 513 Betti number, 133, 135, 137, 140, 149, 342, 513 graded, 137, 140 Bezout, 432 theorem of, 483 bidegree, 457 bihomogeneous, 457 bije tive, 20, 21 bilinear map, 168 binomial, 533

570

Index

birational, 464 blo k, 528 blow{up algebra, 233 blowing up, 532 blowup0, 532 boolean expression, 507 break, 528 breakpoint, 528 Brieskorn, VII Brill{Noether algorithm, 530 brnoeth.lib, 530 browser, 517 Bu hberger, 44, 54, 409 algorithm, 49 fa torizing, 266, 515 Hilbert{driven, 284 spe ialized, 545

riterion, 54, 124, 126, 144, 335

anoni al basis, 93

hart, 445 Castelnuovo{Mumford regularity, 538 Cau hy sequen e, 317 Cayley ubi , 405 Cayley{Hamilton Theorem, 194

hain

riterion, 145, 545 of prime ideals, 207 length, 207 maximal, 208

har, 513

hara teristi polynomial, 162, 163 series, 513

har series, 513

harstr, 513

hart, 438, 445 aÆne, 438 Chevalley, 421 Chinese remainder theorem, 30

lassi ation of singularities, 475, 493, 531

lassify, 531

lassify.lib, 493, 531

leardenom, 513

Clebs h ubi , 405

lose, 513

losed, 414 embedding, 419 immersion, 419, 449 lo ally, 414 subvariety, 414

losure of image, 71, 410, 422 annihilator stru ture, 430 s heme stru ture, 430 proje tive, 432, 441, 442 CoCoA, I

oding, 530

oef, 513

oeÆ ient, 513 leading, 10 matrix, 188 extended, 188

oeffMat, 191

oeffMatExt, 191

oeffs, 513 Cohen, 330 Cohen{Ma aulay maximal, 384 module, 384 ring, 384, 389, 390 test for, 386 CohenMa aulayTest, 387 CohenMa aulayTest1, 395

ohom, 98

okernel, 100

omplete interse tion, 387, 494, 543 ideal, 287 ring, 386 lo al ring, 318, 327, 328

ompletely redu ed, 45

ompletion of a lo al ring, 327 of a module, 327, 329 of lo alization, 330

Index

preserves exa tness, 329

omplex, 128 analyti fun tion, 484 exa t, 128

omponent fun tion, 450

omprehensive Grobner basis, 52, 127

on at, 536

ondu tor, 227

one, 436 aÆne, 435 proje tive, 436 tangent, 479

onstru tible, 421

ontentSB, 127

ontinue, 528

ontra t, 513

ontra tion, 198, 513

ontrahom, 97

ontrol stru tures, 528

onvergent sequen e, 317 Conway polynomial, 6

oordinate ring, 416, 427 homogeneous, 435, 449

orresponden e, 459

ounting nodes, 489

riterion for niteness, 196 for atness, 348, 349 for integral dependen e, 195 for normality, 228, 312

ubi , 413 twisted, 397, 468

upprodu t, 535

y li module, 101 dbprint, 529 de omp,

257

de omp, 271 def, 506 defined, 513

de nition set, 451, 464 deform, 543 deform.lib, 495, 531 deformation, 494

571

equisingular, 532 of singularities, 495 theory, 404, 494 deg, 514 degBound, 529 degree, 2, 114, 122, 285, 287 geometri interpretation, 476 of graded module, 285 of proje tive variety, 475, 514 ordering, 14, 18 global, 14 lo al, 14, 297 weighted, 13, 522 degree, 290, 514 dehomogenization, 438 delete, 514 delete obje ts, 518 delta, 534 depth, 372 depth, 386 determina y, 490, 491 determinantal tri k, 194 develop, 534 diagonal form, 153, 154 diagonalForm, 154 Di kson basis, 12 Di kson's Lemma, 12, 119 diff, 30, 514 dim, 211, 290, 514 dimension, 35, 207, 307, 514, 543

omputation of, 35, 211 embedding, 301 global, 400 of a module, 302, 307 of a monomial ideal, 221 of a proje tive variety, 439 of an aÆne algebra, 218 of an aÆne variety, 290, 416 of an ideal, 207 of bres, 363 proje tive, 393 dire t produ t of modules, 101 dire t sum

572

Index

of modules, 101 of rings, 8 dis riminant, 223, 288, 310, 477, 496, 544 divisible, 119 division theorem Grauert, 333 Weierstra, 320 with remainder, 44 division, 68, 69, 178, 514 domain integral, 23 prin ipal ideal, 23 dominant, 419 download, 497 dual module, 93 dump, 514 early termination, 63 e art, 122, 545 weighted, 56 e art, 545 e ho, 529 elementary divisor, 158 elim.lib, 83, 446, 532 eliminate, 71, 74, 514 elimination, 69, 532 geometri meaning, 422, 461, 469 of omponents, 179 ordering, 40, 70 property, 40 theory main theorem, 460 else, 528 embedding, 419, 449 dimension, 301, 332 Segre, 457 Veronese, 449 endomorphism, 92 Endra, 407 equality test, 25 equidim, 540 equidimensional

de omposition, 263 ideal, 241, 260 part, 260 {th, 262 ring, 260 variety, 471 equidimensional, 261 equidimensional, 273

equidimensionalDe omp,

equising.lib, 532

262

equisingularity stratum, 532 ERROR, 514 Eu lidean algorithm, 66 topology, 471, 479, 484 eval, 514 evaluation, 514 prevent from, 523 exa t sequen e, 128 short, 128 example, 515 exe ute, 503, 515 exit, 500, 515 export, 528 Ext, 535 Ext{ modules, 342 sequen e, 342 extendedDiagonalForm, 160 extension nite, 193 integral, 193 exterior power, 527 extg d, 515 extra weight ve tor, 70, 71 Fa std,

266

fa std, 266, 515

fa tor module, 100 ring, 22 fa torial, 533 fa torize, 6, 25, 515 fa torizing Grobner basis algorithm, 266, 515

Index

faithfully at, 357, 358, 384, 486 fat point, 409, 427 Faugere, 515 Fermat's problem, 404 fet h, 7{9, 515 fet hall, 542 fglm, 515 FGLM algorithm, 515 fglmquot, 515 fibona

i, 533 bre in at family, 363, 364 produ t, 428, 459 eld, 1 of fra tions, 32 perfe t, 9 quotient, 32 residue, 31 ltration, 292 stable, 293 find, 516 nding roots, 546 finduni, 265, 516 nite determina y of singularities, 490 of standard bases, 59 extension, 193 is integral, 194 eld, 5 presentation, 103 nitely determined, 490 generated, 19, 101 niteness test, 324 finitenessTest, 235 finvar.lib, 532 fitting, 345 Fitting ideal, 185, 343, 351 and atness, 351

omputation of, 345 is ompatible with base hange, 344

at, 348, 494

573

faithfully, 357, 358, 384 lo us, 352, 355, 356 is open, 353 flatLo us, 355 flatLo us1, 356

atness and dimension, 363 and standard bases, 364

riterion for, 348, 349 lo al riteria, 359 test for, 354

attening strati ation, 351, 352 stratum, 351 flatteningStrat, 352

oating point numbers, 6 for, 528 formal power series, 315 free lo ally, 346 module, 102, 516 resolution, 133, 393, 544 homogeneous, 136, 517, 519 minimal, 133, 135, 137, 524 periodi , 398 freemodule, 516 fun tion eld, 34 regular, 416 Gau{Manin onne tion, 533 Gaussian elimination, 66, 99 normal form, 536 gaussman.lib, 533 gauss nf, 536 g d, 516 Gelfand, 520, 526 gen, 98, 516 general Ja obian riterion, 307 Noether normalization, 213, 326 position, 247 xn {, 319 general.lib, 533

574

Index

generator, 101 genus, 538 germ, 471 analyti , 484 getdump, 516 Gianni, 246, 268, 271, 515, 540 global dimension, 400 ordering, 11, 13, 118 versus lo al, 35, 470 going down, 205, 225, 363 going up, 225 geometri al meaning, 199 property, 199 graded K {algebra, 114 Betti number, 137 homomorphism, 115 ideal, 115 module, 115 ring, 114 submodule, 115 graph, 421, 429, 459 graphi s, 534, 544 graphi s.lib, 534 Grauert, 228, 333, 494 division theorem, 333 Grobner, 44 basis, 45, 49, 119

omprehensive, 52, 127 redu ed, 46 groebner, 5, 516 Grothendie k, 423 ground eld, 2

omplex numbers, 6 nite, 5 rational numbers, 5 real numbers, 6 with parameters, 6 ring, 2 Gusein-Zade, 407 Hamburger{Noether development, 534

height, 207 and atness, 363 help, 500, 506, 516 Hensel's Lemma, 326 high orner, 60, 491, 517 highest orner, 60, 491 hilb, 280, 282, 283, 517 Hilbert basis theorem, 21 fun tion, 277, 289, 544 aÆne, 291 polynomial, 279, 285, 299 of a hypersurfa e, 285 series, 517 rst, 279, 517 se ond, 279, 517 weighted rst, 280 Hilbert's Nullstellensatz, 31, 219, 415 abstra t, 424 weak form, 218 Syzygy Theorem, 146, 399 Hilbert{driven, 284 Hilbert{Poin are series, 277, 282, 299 Hilbert{Samuel fun tion, 296 multipli ity, 296 polynomial, 296 HilbertPoin are, 284 hilbPoly, 290, 539 Hironaka, 333 hnoether.lib, 534 Hom, 535 homog, 116, 443, 517 homogeneity test, 517 homogeneous

omponent, 114, 434

oordinate ring, 435

oordinates, 433 de omposition, 434 element, 114 free resolution, 136 homomorphism, 115

Index

ideal, 115, 245, 434 prime, 117 part, 115 polynomial, 433 spe trum, 443 submodule, 115 homogenization, 56, 63, 438, 441 weighted, 64, 245, 358 homogenizing variable, 438 homolog.lib, 97, 98, 356, 535 homologi al algebra, 535 homomorphism, 3 lo al, 319 of algebras, 92 of modules, 92 hres, 517 hyperplane at in nity, 438, 441 hypersurfa e aÆne, 413 Hilbert polynomial of, 285 proje tive, 434 ideal, 19 equidimensional, 241, 260 equidimensional part, 260 leading, 41 maximal, 23 membership, 67 perfe t, 398 prime, 23 quotient, 27, 80, 515 geometri interpretation, 82 radi al, 27 redu ed, 27 saturated, 445 vanishing, 415, 424, 435 ideal, 506 identity, 1, 517 I {depth, 372 if, 528 image, 20, 99

losure of, 71, 410, 422 annihilator stru ture, 430 s heme stru ture, 430

omputation of, 464

575

of proje tive morphism, 462 is losed, 461 imaginary part, 518 imap, 7{9, 517 imapall, 542 immersion, 419, 449 impart, 518 impli it fun tion theorem, 325 impli itization, 204 independent set, 219 geometri al meaning, 220 maximal, 219 indepSet, 220, 518 indetermina y set, 464 index of nilpoten y, 27, 28 initial form, 298, 300 ideal, 298{300 inje tive, 20 inout.lib, 95, 104, 535 insert, 518 inSubring, 87, 530 int, 507 integer programming, 535 integral, 193

losure, 201, 203 of an ideal, 202, 541, 542 strong, 201 weak, 201 dependen e, 195 properties of, 199 domain, 9, 23, 28, 29, 306 element, 193 extension, 193 strongly, 201 integrally losed, 203 interpolation, 543 interred, 154, 518 interredu ed, 45, 120 interredu tion, 45 interse t, 79, 102, 518 interse tion, 179 multipli ity, 480{482 of aÆne urves, 482

576

Index

of proje tive urves, 482 of ideals, 79 of s hemes, 427 of submodules, 180 with subrings, 69 intmat, 507 intprog.lib, 535 intStrategy, 127 intve , 508 invariant ring, 530, 532, 542 subspa e, 163 invariant ring, 532 invers, 316 inverse, 26 of power series, 316 inverse, 536 irredu ible, 408 algebrai set, 416, 434

omponent, 408, 416, 437 de omposition, 416, 437, 442 element, 29 polynomial, 4, 8 proje tive variety, 434 topologi al spa e, 425 irredundant, 241 irrelevant ideal, 435, 443 is bije tive, 21, 420, 530 isCohenMa aulay, 392 isCohenMa aulay1, 394 isFlat, 354 is inje tive, 20, 420, 530 isIsolatedSingularity, 239 isLo allyFree, 347 isolated

riti al point, 487 singularity, 487 isomorphi , 92 isomorphism, 20, 417, 427, 449 of modules, 92 proje tive, 433 isReg, 372 is surje tive, 21, 420, 530 ja ob, 30, 518

Ja obian

riterion, 304, 333, 474 general, 307 ideal, 229, 518 matrix, 229, 308 Ja obson radi al, 37, 110 jet, 316, 490, 518 jet, 518 Jordan normal form, 163, 536 jordanform, 536 Kapranov, 520, 526 kbase, 518 k{determined, 490 keepring, 528 Kemper, 540, 547 kernel, 20, 84, 99 kill, 518 killattrib, 518 k{jet, 316, 490, 518 Koszul

omplex, 373, 378 homology, 380 relations, 518 koszul, 518 KoszulHomology, 380 KoszulMap, 380 Kri k, 263, 274, 540 Krone ker symbol, 96 Krull dimension, 207 Krull's interse tion theorem, 110 prin ipal ideal theorem, 218, 303, 304, 482 Kurke, VII La S ala, 519 Laguerre, 76, 518 laguerre, 518 latex.lib, 535 Lazard, 76, 515, 543, 545 l m, 49 lead, 10, 518 lead oef, 10, 518 leadexp, 10, 519

Index

leading

oeÆ ient, 10, 41, 119 exponent, 10, 41 ideal, 41 module, 119 monomial, 10, 41, 119, 545 submodule, 119 term, 10, 41, 119 leadmonom, 10, 519 least ommon multiple, 49, 545 length, 207 of a resolution, 133, 393 lexi ographi al ordering, 13 degree, 13 degree reverse, 13 negative, 14 LIB, 519 library, 504, 519, 530 lift, 26, 53, 68, 178, 519 liftstd, 519 linalg.lib, 163, 536 linear, 92 algebra, 536 link, 508 listvar, 519 LLL.lib, 536 lo al, 484 degree ordering, 297 homomorphism, 319 ordering, 11, 14, 118 ring, 31 algebrai , 484, 486 analyti , 484, 486 at p, 471, 473 regular, 304, 329 lo alization, 32 at a prime ideal, 33

omputing in, 412 of a module, 111 w.r.t. ordering, 39 lo ally

losed, 351, 414 free, 346 Logar, 263, 274, 540

577

long exa t Ext{sequen e, 342 Tor{sequen e, 337 lres, 135, 519 lying over, 225 geometri al meaning, 199 property, 199 Ma aulay, 44, 520, 535 Ma aulay2, I Main Theorem of Elimination Theory, 460 map de nition of, 8 proje tion, 428 quotient, 23 residue, 23 map, 7, 509 mapall, 542 mapIsFinite, 198, 324, 530 Maple, 546 Mathemati a, 534, 550 matrix fa torization, 398 operations, 94 ordering, 12 matrix, 509 matrix.lib, 94, 163, 172, 381, 536 maxideal, 519 maximal

hain, 208 Cohen{Ma aulay, 384 ideal, 23 in general position, 247 i{th power, 519 spe trum, 423 maximality ondition, 22 memory, 519 memory management, 519 Milnor, 505 milnor, 488, 506, 543 Milnor number, 487, 488, 543 total, 487, 504 minAssChar, 540 minAssGTZ, 225, 309, 443, 540

578

Index

minbase, 519

minimal polynomial, 4{6 presentation, 109 resolution, 133, 135 length, 390 standard basis, 45, 120 system of generators, 109, 519, 520 miniversal deformation, 494, 531 minor, 343 minor, 520 minpoly, 529 minres, 135, 150, 520 mixed ordering, 11, 118 modality, 492 module, 91

y li , 101 dual, 93 elimination of omponents, 180 nitely generated, 101 free, 102, 516 homomorphism, 92 isomorphi , 92 Noetherian, 108 of dierentials, 314, 336 of nite presentation, 103 of syzygies, 140 ordering, 118 quotient, 183 shift of, 115 sparse representation, 104 support, 112 twist of, 115 module, 509 module ontainment, 530 modulo, 107, 187, 340, 380, 520 mod versal, 531 Moller, 76, 543 mondromy.lib, 537 monitor, 520 monodromy, 533, 537 monodromy, 533 monomial, 2, 118

leading, 10 ordering, 10, 118 module, 118

MonomialHilbertPoin are, 281

Mora, 44, 55, 515 normal form, 57, 123, 545 morphism, 3, 417, 427, 448, 454 dominant, 419 proje tive, 460 Morse lemma, 489, 490 generalized, 493 mpresmat, 520 mprimde .lib, 245, 537 mregular.lib, 538 mres, 135, 150, 394, 520 M {sequen e, 371 mstd, 520 mult, 520 multBound, 529 multipli ative subset, 32 multipli atively losed, 32 multipli ity, 297, 332, 475, 479, 521 geometri interpretation, 479 Hilbert{Samuel, 296 interse tion, 482 of monomial ideal, 520 w.r.t. an ideal, 296 multsequen e, 534 MuPAD, 551 Nakayama's Lemma, 108, 117

nameof, 521 names, 521 n ols, 521

Newton's Lemma, 326 NFBu hberger, 49, 50, 52, 55, 57, 122, 126 NFMora, 57, 68, 122, 123, 126, 545 nilpotent, 27, 28 index, 28 nilradi al, 27 node, 489 Noether, 223 normalization, 213, 324, 479 for lo al rings, 217

Index

general, 213, 326 over nite elds, 216 theorem, 213 noether, 529 Noetherian module, 108 ring, 21 NoetherNormal, 233 noetherNormal, 233, 393, 530 NoetherNormalization, 215 non{normal lo us, 227 is losed, 227 non{zerodivisor, 23 non{normalLo us, 231 normal, 203

one, 300 form, 46, 120, 334 Gaussian, 536 Jordan, 163, 536 over rings, 52 polynomial, 47, 120, 126 pseudo, 126 rational, 163, 167 redu ed, 46, 66, 120 w.r.t. an ideal, 48 weak, 47, 120 without division, 126 ring, 203 variety, 411 normal, 202, 230, 236, 538 normal.lib, 202, 207, 538 normalI, 542 normality riterion, 228 normalization, 203, 538 is nite, 223 normalization, 230 normalization, 236 npars, 521 nres, 521 nrows, 521 ntsolve.lib, 538 Nullstellensatz, 423 abstra t, 424 Hilbert's, 217

579

proje tive, 436 number of parameters, 521 of points, 265 of variables, 521 number, 5 nvars, 521 open, 521 option, 521 option(redSB), 46, 66, 522

ord, 315 order, 297, 315 weighted, 315 ordering blo k, 14 degree, 14, 18 degree lexi ographi al, 13 degree reverse lexi ographi al, 13 elimination, 40 global, 11, 118 degree, 14 lexi ographi al, 13 lo al, 11, 14, 118 matrix, 12 mixed, 11, 118 monomial, 10 negative degree lexi ographi al, 14 negative degree reverse lexi ographi al, 14 negative lexi ographi al, 14 produ t, 14 semigroup, 10 test, 542 weighted degree, 14 weighted lexi ographi al, 13 weighted reverse lexi ographi al, 13 ordstr, 522 output format, 522 par, 522 paramet.lib, 539

580

Index

parameter, 6, 7, 34 parametrization, 201, 539 parametrize, 539 pardeg, 522 parstr, 522 part at in nity, 440 partial derivative, 514 perfe t eld, 9, 213, 247 ideal, 398 periodi free resolution, 398 Pinkham, 495 plane urve, 413 plot, 197, 407, 545 pmat, 94 Poin are omplex, VI, 336 point

losed, 425 non{singular, 474 of Pn , 433 regular, 474 singular, 474 pole set, 451, 464 poly, 510 poly.lib, 290, 539 polynomial

onstant, 2 degree of, 2 distributive representation, 3 fun tion, 3, 8 irredu ible, 4 minimal, 4, 6 normal form, 47 over A, 2 re ursive representation, 3 ring, 7 is Noetherian, 21 weak normal form, 120 power series, 315 expansion, 41, 119 ring, 315 is omplete, 318 is lo al, 316 is Noetherian, 322

P -primary, 241

preimage, 19, 21 preimage, 20, 21, 85, 522 prepareQuotientring, 269 prepareSat, 271 presentation matrix, 103 minimal, 109 presolve.lib, 539 primary de omposition, 241, 409, 537, 540 irredundant, 241 of a submodule, 246 ideal, 241 submodule, 246 primaryTest, 251 primaryTest, 267 primde .lib, 25, 27, 78, 276, 309, 356, 395, 437, 443, 540 primde GTZ, 25, 540 primde SY, 25, 540 prime avoidan e, 24 element, 29 ideal, 23, 33 embedded, 241 minimal asso iated, 208 relevant, 443 prime, 522 primes, 533 primitiv.lib, 541 primitive element, 9, 541 prin ipal ideal, 19 domain, 23, 153, 307 ring, 23 open set, 444, 450 print, 94, 95, 522 printlevel, 529 pro , 510 produ t

riterion, 63, 145, 545 of Pn , 457

Index

of ideals, 19 of varieties, 421 universal property, 458 ordering, 14 projdim, 394 proje tion from H , 468 from a point, 455 onto linear subspa e, 410 proje tive, 357 algebrai set, 434

losure, 432, 441, 442

one, 436 dimension, 393, 394 hypersurfa e in Pn , 434 isomorphism, 433 morphism, 460 image of, 461, 462 n{spa e, 433 Nullstellensatz, 436 resolution, 394 s heme, 446 s heme over R, 445 spa e, 433 spa e over R, 444 spe trum, 443 subs heme of PnR , 445 variety, 434 dimension of, 439 isomorphism, 449 morphism, 448 proje tively equivalent, 450 prompt, 505 prune, 109, 340, 341, 380, 523 pseudo normal form, 126 polynomial, 126 standard basis, 52, 127 Puiseux development, 534 pure dimensional, 471 ideal, 241, 260 ring, 260 qhmoduli.lib, 541 qhweight, 116, 523

581

qring, 25, 511 quadri , 413 quarti , 413 rational, 397 quasi{aÆne, 414 quasi{proje tive s heme, 445, 447 variety, 434, 447 aÆne, 449 quasihomogeneous, 114, 336, 541, 544 qui k lass, 493, 531 quinti , 413 quit, 500, 528 quote, 523 quotient, 101, 544 eld, 32, 33 map, 23 module, 100 of ideals, 27, 81 of module by ideal, 101, 102, 184 of submodules, 101 ring, 22, 25, 511 total, 33 quotient, 27, 102, 182, 184, 523

(R1), 312 (Ri), 390 Rabinowi h's tri k, 78 radi al, 27, 183, 192, 265 Ja obson, 37, 110 membership, 77, 78, 183 radi al, 264 radi al, 27, 78, 274, 309, 437, 540 random, 523 random.lib, 478, 541 rank, 102, 343 rational fun tion, 451 map, 464 normal urve, 289, 478 normal form, 163, 167 numbers, 5 read, 503, 523

582

Index

read from a le, 503 real, 6 part, 524 reddevelop, 534

redNFBu hberger,

redu e, 25, 51, 523

50

redu ed, 45

ompletely, 45 element, 120 ideal, 27 normal form, 46, 66, 120, 334 is unique, 48, 121 ring, 27, 29, 390 s heme, 445 set, 45, 120 redu ible, 425 redu tion of a ring, 27, 210 of a set, 45 redu tionToZero, 256 Rees{Algebra, 232, 541 ReesAlgebra, 542 rees los.lib, 541 regular at a point, 448 fun tion, 416, 448 germ of, 471 lo al ring, 304, 307, 329, 400, 474 homologi al hara terization, 399 is Cohen{Ma aulay, 384 is integral domain, 306 is normal, 307, 312 test for, 400 point, 474 ring, 311, 314, 400 sequen e, 306, 371, 372 system of parameters, 303 variety, 474 xn {, 319 regularity, 524 relation, 140 trivial, 19

relevant prime ideal, 443 Remmert, 228 repart, 524 representation of polynomial distributive, 3 re ursive, 3 representative, 100 res, 135, 394, 524 reservedName, 524 residue eld, 31 map, 23 Resolution, 142 resolution, 133, 136, 524, 544 homogeneous, 136, 517, 519 length of, 393 minimal, 133, 135 periodi , 398 proje tive, 394 S hreyer, 150 resolution, 511 resultant, 431, 524 multipolynomial, 520, 543 resultant, 431, 524 return, 504, 528 Riemann, 492 singularity removable theorem, 410 right equivalent, 490 right exa t, 171 ring, 1, 7 aÆne, 23 Artinian, 30 asso iated to K [x℄ and >, 39, 40

ommutative, 1

omplete interse tion, 386

oordinate, 416 default, 504 de nition of, 503 dire t sum, 8 fa tor, 22 map, 3 Noetherian, 21

Index

of fra tions, 32 total, 33 polynomial, 2 prin ipal ideal, 23 quotient, 22, 25, 511 total, 33 redu ed, 27 redu tion of, 27 regular, 311 semi{lo al, 31 ring, 504, 511 ring.lib, 542 rinvar.lib, 542 Robbiano, 13 RowNF, 154 rvar, 524 (S2), 312 (Si), 390

sameComponent, 545 Samuel, 521 sat, 83, 446, 532 saturated, 445, 462 saturation, 27, 37, 81, 83, 445, 446, 532 exponent, 81 geometri interpretation, 82, 445 s alar multipli ation, 91 S hanuel's Lemma, 152 s heme, 445 aÆne, 427 interse tion, 427 proje tive, 443 stru ture, 444, 445 union, 427 S hreyer, 143 ordering, 143 resolution, 150 Segre embedding, 457 threefold, 458 semi{lo al, 31 semi{universal deformation, 494 separable

583

eld extension, 9 polynomial, 9 Serre, 307, 312, 399

onditions, 312, 390 setring, 7, 524 shift of a module, 115 Shimoyama, 537, 540 short, 7, 529 short exa t sequen e, 128 show, 104 simplify, 524 sing.lib, 480, 488, 543 singular at a point, 474 lo us, 308, 309, 358, 474, 475, 543 point, 474 singularity, 411, 485 ADE, 492

lassi ation of, 493 hyperboli , 493 isolated, 487 Kleinian, 492 paraboli , 492 simple, 492 spe trum, 543, 544 singularLo us, 309 singularLo usEqui, 308 size, 525 slo us, 543 snake lemma, 131 solvability, 74 solve, 77, 543 solve.lib, 76, 543 solve IP, 535 solving, 75, 76, 187

omplex, 543 linear equations, 187 Newton, 538 prepare for, 539 with polynomial onstraints, 189 sortve , 525 spa e urve, 543

584

Index

sparsepoly, 478 sparsetriag, 216 sp urve.lib, 543

spe ialization of a standard basis, 52 spe trum, 543, 544 homogeneous, 443 maximal, 423 prime, 423 proje tive, 443 spe trum.lib, 544 splitting lemma, 493 spoly, 545 s{polynomial, 49, 122 squarefree, 254, 534 part, 274 squarefree, 274 sres, 135, 150, 151, 525 SResolution, 147 stable ltration, 293 Standard, 54, 122 standard basis, 45, 52, 119, 333, 526, 544, 545 nite determina y, 59 in power series ring, 333 minimal, 45, 120 over a ring, 52, 67 pseudo, 52, 127 spe ialization of, 52, 125 without division, 127 representation, 46, 120 standard, 545 standard.lib, 544 StandardBasis, 59, 123 status, 525 std, 5, 284, 526 stdfglm, 526 stdhilb, 526 strati ation theory, 404 stratify.lib, 544 string, 502, 511 strong integral losure, 201 strongly integral, 201

Sturmfels, 407 subalgebra membership, 86 submodule, 98 subring, 1 subs heme, 427 subst, 526 substitution, 318 substring, 516 subvariety, 414

omplement of, 410 of Pn Pm, 457 sum of submodules, 101 support, 112, 185 surf, 407, 497, 498, 544 surf.lib, 197, 204, 407, 413, 544 surfa e, 413 in 3-spa e, 413 surje tive, 20, 21 swallow tail, 496 system of generators, 19 minimal, 109 parameters, 303 syz, 141 syz, 99, 141, 526 syzygy, 140 module, 140, 409 theorem, 146 tail, 10, 41, 119, 545 tail, 545 tangent one, 298, 479, 543 tangent one, 480, 543 tea hstd.lib, 545 tensor, 537 tensor produ t, 167, 537 is right exa t, 171 of maps, 170 of modules, 168, 172 of rings, 175, 542 universal property, 174, 428 tensorMod, 172 term, 2 leading, 10 test ideal for normality, 228

Index tjurina, 488, 543

Tjurina number, 336, 487, 488, 543 total, 487 Togliatti quinti , 413 topology m{adi , 317 Eu lidean, 479, 484 Zariski, 23, 414, 424, 434, 444 Tor, 337

omputation of, 340 Tor, 340 Tor{ module, 337 sequen e, 337 tori ideal, 545 tori .lib, 545 torsion free, 101 module, 101, 102 submodule, 101 total Milnor number, 487 ring of fra tions, 33 Tjurina number, 487 TRACE, 529 tra e, 526 Trager, 246, 268, 271, 540 transpose, 526 triang.lib, 76, 545 triangMH, 546 triangular set, 76, 543, 545 trivial relation, 19 twist of a module, 115 twisted ubi , 468 type, 526 typeof, 526 unique fa torization domain, 38, 206 unit, 1 universal property of bre produ t, 428 of lo alization, 34 of produ t, 458 of tensor produ t, 174

585

universally Japanese, 223 vandermonde, 527

Vandermonde system, 527 vanishing ideal, 415, 424, 435 var, 527 variety aÆne algebrai , 412 proje tive, 431, 443 varstr, 527 Vas on elos, 233 vdim, 77, 478, 527 ve tor, 98 ve tor, 511 Veronese embedding, 449, 476 d{tuple, 449 variety, 476 versal, 495, 531 w{deg, 13, 370 w{ord, 315 Wall, 492 weak normal form, 47, 120 wedge, 527 weierstr.lib, 546 Weierstrass, 321 Weierstra, 546 division theorem, 320 polynomial, 217, 321 preparation theorem, 321, 479 semigroup, 530 weight, 527 weight{ve tor, 15, 315 weighted degree, 13, 114, 315 ordering, 14 e art, 56 homogeneous, 114 order, 315 Weispfenning, 52 while, 528 Whitney umbrella, 74, 230, 231, 404, 406 write, 503, 527

586

Index

write to a le, 503 Yokoyama, 537, 540 Za harias, 246, 268, 271, 540 Zariski

losed, 414

losure of image, 74

onje ture, VI topology, 23, 414, 424, 434, 444 basis, 444 Zelevinsky, 520, 526 zero{dimensional ideal in general position, 247 zero{set, 412, 423, 434, 443 number of points, 265 zeroDe omp, 252 zeroDe omp, 268 zerodivisor, 23, 183 test, 25, 184 zeroradi al, 264 zeroset.lib, 546 Zorn's Lemma, 31

Algorithms Algorithms

basisElement, 379 basisNumber, 378

oeMat, 191

oeMatExt, 191 CohenMa aulayTest, 387 CohenMa aulayTest1, 395 de omp, 257, 271 depth, 386 diagonalForm, 154 equidimensional, 261, 273 equidimensionalDe omp, 262 extendedDiagonalForm, 160 Fa std, 266 nitenessTest, 235 tting, 345

atLo us, 355

atLo us1, 356

atteningStrat, 352 HilbertPoin are, 284 invers, 316 isCohenMa aulay, 392 isCohenMa aulay1, 394 isFlat, 354 isLo allyFree, 347 isReg, 372 KoszulHomology, 380 KoszulMap, 380 mapIsFinite, 324 maple fa torize, 549 Milnor, 505 MonomialHilbertPoin are, 281, 282 mupad fa torize, 553 NFBu hberger, 49 NFMora, 57, 123

NoetherNormal, 233 NoetherNormalization, 215 non{normalLo us, 231 normalization, 230, 236 polyOfEndo, 165 prepareQuotientring, 269 prepareSat, 271 primaryTest, 251, 267 projdim, 394 radi al, 264, 274 redNFBu hberger, 50 redu tionToZero, 256 Resolution, 142 RowNF, 154 singularLo us, 309 singularLo usEqui, 308 squarefree, 274 SResolution, 147 Standard, 54, 122 StandardBasis, 59, 123 syz, 141 tensorMod, 172 Tor, 340 Weierstrass, 321 zeroDe omp, 252, 268 zeroradi al, 264

587

588

Algorithms

SINGULAR{Examples

589

SINGULAR{Examples

algebrai dependen e, 87 annihilator, 186 Betti numbers, 135 graded, 137

lassi ation of singularities, 493

omputation in elds, 5 in polynomial rings, 7 in quotient rings, 25 of d(I; K [x℄), 222 of Hom, 106 of the dimension, 211 of Tor, 340

omputing with radi als, 27

ounting nodes, 489

reating ring maps, 8

y li de omposition, 159 deformation of singularities, 495 degree, 289 of proje tion, 469 of proje tive variety, 478 diagonal form, 154 dimension, 289 embedding, 304 of a module, 302 elimination and resultant, 431 of module omponents, 180 of variables, 71 proje tive, 466 equidimensional de omposition, 263 part, 261 estimating the determina y, 491 nite maps, 196 niteness test, 324 Fitting ideal, 186

Fitting ideals, 345

at lo us, 356

atness test, 369

attening strati ation, 352 global versus lo al rings, 35 graded Betti numbers, 137 rings and modules, 116 highest orner, 60 Hilbert fun tion, 289 Hilbert polynomial, 299 Hilbert{Poin are series, 282 homogeneous resolution, 137 ideal membership, 68 image of module homomorphism, 99 independent set, 220 initial ideal, 299 inje tive, 420 integral

losure of an ideal, 202 elements, 195 interse tion of ideals, 79 of submodules, 102, 181 inverse of a power series, 316 Ja obian

riterion, 304 Jordan normal form, 163 kernel of a ring map, 85 of module homomorphism, 99, 187 Koszul omplex, 378 leading data, 11

590

SINGULAR{Examples

linear ombination of ideal members, 68 lo al and global dimension, 472 lying over theorem, 225 maps indu ed by Hom, 96 matrix operations, 94 Milnor and Tjurina number, 488 minimal asso iated primes, 209 presentations, 109 module annihilator, 186 membership, 178 presentation of, 104 quotient, 102 radi al and zerodivisors, 184 monomial orderings, 16 morphisms of proje tive varieties, 455 multipli ity, 480 Noether normalization, 216 non{normal lo us, 232 normal form, 51, 123 normalization, 230 Poin are series, 299 presentation of a module, 104 primary de omposition, 258 test, 252 proje tive

losure, 443 elimination, 466 Nullstellensatz, 437 subs hemes, 446 properties of ring maps, 20 quotient of ideals, 81 of submodules, 102, 183 radi al, 184, 265 membership, 78 realization of rings, 42

redu tion to zero{dimensional ase, 257 regular sequen es, 372 system of parameters, 304 regularity test, 400 resolution, 135 homogeneous, 137 saturation, 83, 446 S hreyer resolution, 150 singular lo us, 309 solving equations, 76 linear, 189 standard bases, 59, 124 subalgebra membership, 87 submodules, 104 interse tion of, 102, 181 of An , 98 sum of submodules, 102 surfa e plot, 407, 413 surje tive, 420 syzygies, 141 tangent one, 480 tensor produ t of maps, 170 of modules, 172 of rings, 175 test for Cohen{Ma aulayness, 386, 392, 394 for atness, 354 for lo al freeness, 347 Weierstra polynomial, 321 Zariski losure of the image, 74 zero{dim primary de omposition, 253 zerodivisors, 184 z {general power series, 320

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