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 di eren e between theory and pra ti e. In pra ti e there is. Yogi Berra
A SINGULAR Introdu tion to Commutative Algebra o ers 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 e e 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 di ers 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 e e 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 di erentials. 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 o ers 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
1. Rings, Ideals and Standard Bases
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 = x1 1 : : : xn n ;
= ( 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 = ax 1 1 : : : x nn ; 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
a 1 ::: n x 1 1 : : : x nn ;
with a 2 A. For 2 N n , let j j := 1 + + n . The integer deg(f ) := maxfj j 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
1. Rings, Ideals and Standard Bases
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 di eren 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.
1.1 Rings, Polynomials and Ring Maps
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 di eren 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)
1.1 Rings, Polynomials and Ring Maps
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 di erent 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
1. Rings, Ideals and Standard Bases
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
1. Rings, Ideals and Standard Bases
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 di erent 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, di erent orderings have di erent 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 .
1.2 Monomial Orderings
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 di erent 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
1. Rings, Ideals and Standard Bases
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 = f 0 = ( 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 M n and, sin e B n is a Di kson basis of M n , there is a 0 2 B n 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 ;
1.2 Monomial Orderings
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 di erent 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 di er: 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.
1.2 Monomial Orderings
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, di erent 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 di eren 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 x 1 > x 1 implies that x 1 + 2 > x 1 + 2 , and x 2 > x 2 implies that x 1 + 2 > x 1 + 2 , therefore x 1 + 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
1. Rings, Ideals and Standard Bases
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 di erent orderings, whi h are onsidered as di erent rings in Singular. Then we map a given polynomial f to the di erent 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. Rings, Ideals and Standard Bases
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
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. Rings, Ideals and Standard Bases
' 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
1. Rings, Ideals and Standard Bases
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℄:
1.3 Ideals and Quotient Rings
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
1. Rings, Ideals and Standard Bases
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 e e tively using Singular. We de ne a quotient ring and test equality and the zerodivisor property in the quotient ring.
1.3 Ideals and Quotient Rings
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 di eren 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 = <0>: 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
1. Rings, Ideals and Standard Bases
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 e e 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 e e 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
1.3 Ideals and Quotient Rings
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 = j j0 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.
1.3 Ideals and Quotient Rings
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 di erent 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 di erent 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. Rings, Ideals and Standard Bases
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 di erent 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
1. Rings, Ideals and Standard Bases
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).
1.4 Lo al Rings and Lo alization
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
1. Rings, Ideals and Standard Bases
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.
1.4 Lo al Rings and Lo alization
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 di eren 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
1. Rings, Ideals and Standard Bases
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 e e 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
1. Rings, Ideals and Standard Bases
(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 e e 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 .
1.5 Rings Asso iated to Monomial Orderings
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
1. Rings, Ideals and Standard Bases
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 <1 y and x <2 y but xy + 1 2 S<1 and xy + 1 62 S>2 .
SINGULAR Example 1.5.10 (realization of rings).
We show how to reate the rings of Examples 1.5.3. Note that Singular sorts the monomials with respe t to the monomial ordering, the greatest being rst. Hen e, the position of 1 in the output shows whi h monomials are greater, respe tively smaller, than 1. int n,m=2,3; ring A1 = 0,(x(1..n),y(1..m)),(dp(n),ds(m)); poly f = x(1)*x(2)^2+1+y(1)^10+x(1)*y(2)^5+y(3); f; //-> x(1)*x(2)^2+x(1)*y(2)^5+1+y(3)+y(1)^10 1>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);
1.5 Rings Asso iated to Monomial Orderings
43
//-> y(1)^10+y(3)+1+x(1)*y(2)^5+x(1)*x(2)^2 x(1)*y(2)^5<1; //-> 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. Rings, Ideals and Standard Bases
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 di erent versions of standard basis algorithms are due to di erent 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
1. Rings, Ideals and Standard Bases
(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).
1.6 Normal Forms and Standard Bases
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).
48
1. Rings, Ideals and Standard Bases
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
1.6 Normal Forms and Standard Bases
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 di erent normal form fun tion.
50
1. Rings, Ideals and Standard Bases
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);
1.6 Normal Forms and Standard Bases
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
52
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
1. Rings, Ideals and Standard Bases
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.
1.7 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 di erent 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 x i 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 x i , 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
1. Rings, Ideals and Standard Bases
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 + j j > q + j j or if p + j j = 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).
1.7 The Standard Basis Algorithm
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
1. Rings, Ideals and Standard Bases
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:
1.7 The Standard Basis Algorithm
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 di erent standard bases with respe t to di erent 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);
1.7 The Standard Basis Algorithm
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
1. Rings, Ideals and Standard Bases
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 (
f^i :=
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.
1.7 The Standard Basis Algorithm
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(f^i ) 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
1. Rings, Ideals and Standard Bases
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 ).
1.7 The Standard Basis Algorithm
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.
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1. Rings, Ideals and Standard Bases
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 x 1 1 : : : xi i 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 .)
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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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1. Rings, Ideals and Standard Bases
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
1.8 Operations on Ideals and their Computation
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 :
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1. Rings, Ideals and Standard Bases
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.
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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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1. Rings, Ideals and Standard Bases
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
1.8 Operations on Ideals and their Computation
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 .
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1. Rings, Ideals and Standard Bases
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.
1.8 Operations on Ideals and their Computation
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) di erential 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
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1. Rings, Ideals and Standard Bases
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
1.8 Operations on Ideals and their Computation
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
1. Rings, Ideals and Standard Bases
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.
1.8 Operations on Ideals and their Computation
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 di erent 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).
1.8 Operations on Ideals and their Computation
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
1. Rings, Ideals and Standard Bases
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 ) :
1.8 Operations on Ideals and their Computation
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
1.8 Operations on Ideals and their Computation
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 di erent 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.
1.8 Operations on Ideals and their Computation
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
1. Rings, Ideals and Standard Bases
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.
1.8 Operations on Ideals and their Computation
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
A. Geometri Ba kground
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 di erent 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 di eren 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
A. Geometri Ba kground
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-
A.1 Introdu tion by Pi tures
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 di erent, 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
A. Geometri Ba kground
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
A.1 Introdu tion by Pi tures
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
A. Geometri Ba kground
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
A.1 Introdu tion by Pi tures
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 di erent 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
A. Geometri Ba kground
(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
A. Geometri Ba kground
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.
A.2 AÆne algebrai varieties
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 di er 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
A. Geometri Ba kground
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.
A.2 AÆne algebrai varieties
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
A. Geometri Ba kground
fe := (fe1 ; : : : ; fem) : A nK
! A mK :
Sin e the possible hoi es for fei di er 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 ℄)
A.2 AÆne algebrai varieties
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
A. Geometri Ba kground
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.
A.2 AÆne algebrai varieties
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. Geometri Ba kground
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
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 e e 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
A. Geometri Ba kground
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.
A.3 Spe trum and AÆne S hemes
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
A. Geometri Ba kground
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.
A.3 Spe trum and AÆne S hemes
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
A. Geometri Ba kground
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:
A.3 Spe trum and AÆne S hemes
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
