GTM256.A.Course.in.Commutative Algebra.pdf交换代数GTM2

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详细说明：交换代数GTM256.A.Course.in.Commutative AlgebraTo Idaleixis and martin Contents Introduction 鲁鲁 Part I The Algebra geometry Lexicon 1 Hilbert's nullstellensatz 17 1.1 Maximal ideals 18 1.2 Jacobson rin 1. 3 Coordinate rings 26 E xercises 2 Noetherian and Artinian R ings 2. 1 The Noether and Artin Property for Rings and Modules 2.2 Noetherian Rings and modules,,,...........,. 38 Exercises 3 The Zariski Topology 43 3.1 Affine varictics 13 3.2 Spectra 46 3.3 Noetherian and Irreducible spaces 48 Exercises 4 A Summary of the lexicon ..55 4.1 True geometry: Affine Varieties.............. 55 4.2 Abstract Geometry: Spectra 56 Exercises .58 Part II Dimension 5 Krull Dimension and Transcendence Degree Exercises 70 Contents 6 Localizati 73 E 80 7 The Principal Ideal Theorell 85 7. 1 Nakayama's Lemma and the Principal Ideal Theorem 7.2 The Dimension of fibers Exercises 番番·,番 8 Integral extensions 103 8.1 Integral Closure 103 8.2 Lying Over, Going Up and going Down 8. 3 Noether normalization 114 Exercises 121 Part I Computational Method 9 Grobner bases 127 9. 1 Buchberger's Algorith 128 9.2 First Application: Elimination Ideals 137 Exercises ,143 10 Fibers and Images of morphisms revisited 147 10.1 The Generic Freeness Lemma 147 10.2 Fiber Dimension and Constructible Sets .152 10.3 Application: Invariant Theory 154 Exercises 158 11 Hilbcrt Scrics and dimension 11. 1 The hilbert-Serre Theorem 161 11.2 Hilbert polynomials and dimension 167 Exercises 171 Part Iv Local Rings 12 Dimension Theory 12. 1 The Length of a Module 177 12.2 The Associated Graded ring 180 E 18 13 Regular Local Rings 191 13. 1 Basic Properties of Regular Local Rings 191 13.2 The Jacobian Criterion 195 E 203 Contents 14 Rings of Dimension One 207 14.1 Regular Rings and normal rings 207 14.2 Multiplicative Ideal theory 211 14.3 Dedekind domains 216 Exercises Solutions of exercises 227 References 309 Notation 313 nae 垂··鲁垂鲁 315 Introduction Colninutative algebra is the theory of coInllutative rings Its historic roots are in invariant, theory, number theory, and, most importantly, geometry. Con sequently, it nowadays provides the algebraic basis for the fields of algebraic number theory and algebraic geometry. Over recent decades, commutative al gebra has also developed a vigorous new branch, computational commutative algebra, whose goal is to open up the theory to algorithmic computation. so rather than being an isolated subject, commutative algebra is at the cross- roads of several important mathematical disciplines This book has grown out of various courses in commutative algebra that I nave taught in Hcidclbcrg and Munich. Its primary objcctivc is to scrvc as a guide for an introductory graduate course of one or two semesters, or for self- study. I have striven to craft a text that presents the concepts at, the center of the field in a coherent, tightly knitted way, with streamlined proofs and a focus on the core results. Needless to say, for an imperfect writer like me, such high-flying goals will always remain elusive. To introduce readers to the more recent algorithmic branch of the subject, one part of the book is devoted to computational mcthods. The connections with gcomctry arc morc than just applications of coIlImlutative algebra to allother InatheInatical field. In fact virtually all concepts and results have natural geometric interpretations that bring out the"true meaning"of the theory. This is why the first part of the book is entitled "The Algebra geometry Lexicon, and why I have tried to keep a focus on the geometric context throughout. Hopefully, this will make the theory more alive for readers, more meaningful, more visual, and easier to remember How To Use the book The main intention of the book is to provide material for an introductory graduate course of one or two semesters. The duration of the course clearly depends on such parameters as speed and teaching hours per week and on 10 Introduction how much material is covered. In the book, I have indicated three options for skipping material. For example, one possibility is to omit Chapter 10 and most of Section 7.2. Another is to skip Chapters 9 through 11 almost entirel But apart from these options, interdependencies in the text are close enough to makc it hard to skip matcrial without tearing holes into proofs that comc later. So the instructor can best lilnlit the amount of Material by choosing where to stop. A relatively short course would stop after Chapter 8, while other natural stopping points are after Chapter 1l or 13 The book contains a total of 143 exercises. Some of them deal with ex- amples that illustrate definitions(such as an example of an Artinian module that is not Noetherian)or shed some light on the necessity of hypotheses of theorems(such as an example where the principal ideal theorem fails for a I1Oll-Noethieriall ring). Others give extensions to the theory(such as a series of exercises that deal with formal power series rings), and yet others invite readers to do computations on examples. Thesc cxamples oftcn comc from geometry and also serve to illustrate the theory (such as examples of desingu larization of curves). Some exercises depend on others, as is usually indicated in the hints for the exercise. However, no theorem, lemma, or corollary in the text depends on results from the exercises. I put a star by some exercises to indicate that i consider them more difficult. Solutions to all exercises are provided on a cd that comes with the book. In fact, the CD contains an clectronic vcrsion of thc cntirc book. with solutions to thc cxcrciscs Although the ideal way of using the book is to read it from the beginning to the end (every author desires such readers! ) an extensive subject index hould facilitate a less linear navigation. In the electronic version of the book all cross-references are realized as hyperlinks, a feature that will appeal to readers who like working on the screen Prerequisites Readers should have taken undergraduate courses in linear algebra and ab stract algebra. Everything that is assumed, is contained in Lang's book [33] but certainly not everything in that book is assumed. Specifically, readers should have a grasp of the following subjects definition of a(commutative) ring, ideals, prime ideals and maximal ideals zero divisors quotient rings (also known as factor rings) subrings and hioinoinorphisins of rings principal ideal domains factorial rings(also known as unique factorization domains polynomial rings in several indeterminates finite field extensions. and algebraically closed fields

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