eBook2nd, revised Edition (2nd, revised Edition)
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Overview
The primary audience for this book is students and the young researchers interested in the core of the discipline. Commutative algebra is by and large a self-contained discipline, which makes it quite dry for the beginner with a basic training in elementary algebra and calculus.
A stable mathematical discipline such as this enshrines a vital number of topics to be learned at an early stage, more or less universally accepted and practiced. Naturally, authors tend to turn these topics into an increasingly short and elegant list of basic facts of the theory. So, the shorter the better. However, there is a subtle watershed between elegance and usefulness, especially if the target is the beginner. From my experience throughout years of teaching, elegance and terseness do not do it, except much later in the carrier. To become useful, the material ought to carry quite a bit of motivation through justification and usefulness pointers.
On the other hand, it is difficult to contemplate these teaching devices in the writing of a short book. I have divided the material in three parts. starting with more elementary sections, then carrying an intermezzo on more difficult themes to make up for a smooth crescendo with additional tools and, finally, the more advanced part, versing on a reasonable chunk of present-day steering of commutative algebra.
Historic notes at the end of each chapter provide insight into the original sources and background information on a particular subject or theorem.
Exercises are provided and propose problems that apply the theory to solve concrete questions (yes, with concrete polynomials, and so forth).
Product Details
| ISBN-13: | 9783111078847 |
|---|---|
| Publisher: | De Gruyter |
| Publication date: | 08/07/2023 |
| Series: | De Gruyter Textbook |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 370 |
| File size: | 50 MB |
| Note: | This product may take a few minutes to download. |
| Age Range: | 18 Years |
About the Author
Aron Simis is Professor Emeritus: Universidade Federal de Pernambuco, Recife, Brazil, and a Class A research scholarship recipient from the Brazilian Research Council.[1] He earned his Ph.D. from Queen's University, Canada. He has previously held a full professorship at IMPA (Instituto de Matemática Pura e Applicada) in Rio de Janeiro, Brazil. He was president of the Brazilian Mathematical Society and a member on several occasions of international commissions of the IMU (International Mathematical Union) and TWAS (Academy of Sciences for the Developing World).[2]
At large he is a John Simon Guggenheim Fellow[4] and has been awarded other fellowships from the Max Planck Institute, Japan Society for Promotion of Science, and the Istituto Nazionale di Alta Matematica. He is a member both of the Brazilian Academy of Sciences and the Academy of Sciences for the Developing World (Trieste, Italy).
His main research interests in mathematics include main structures in commutative algebra; projective varieties in algebraic geometry; aspects of algebraic combinatorics; special graded algebras; foundations of Rees algebras; cremona and birational maps; algebraic vector fields; differential methods.
Table of Contents
Thanks v
Foreword vii
Part I
1 Basic introductory theory 3
1.1 Commutative rings and ideals 3
1.1.1 Ideals, generators 3
1.1.2 Ideals, residue classes 4
1.1.3 Ideal operations 5
1.1.4 Prime and primary ideals 8
1.1.5 A source of examples: monomial ideals 9
1.2 Algebras 11
1.2.1 Polynomials and finitely generated algebras 11
1.2.2 The transcendence degree 12
1.2.3 Basic properties of the transcendence degree 15
1.3 Historic note 17
1.3.1 Terminology 17
1.3.2 Early roots 17
1.4 Exercises 19
2 Main tools 23
2.1 Rings of fractions 23
2.1.1 Definitions 23
2.1.2 General properties of fractions 24
2.1.3 Local rings and symbolic powers 27
2.2 Integral ring extensions 28
2.2.1 Preliminaries 28
2.2.2 The Cohen-Seidenberg theorems 30
2.2.3 Integral closure of ideals 32
2.3 Krull dimension and Noether normalization 35
2.3.1 Behavior in integral extensions 36
2.3.2 Noether normalization and the dimension theorem 36
2.3.3 Complements to Noether's theorem 37
2.4 Nullstellensatz 38
2.5 Dimension theory I 42
2.5.1 Noetherian and Artinian rings 42
2.5.2 Associated primes 50
2.5.3 Krull's principal ideal theorem 53
2.5.4 Dimension under extensions 55
2.6 Primary decomposition 60
2.6.1 The nature of the components 60
2.6.2 The Lasker-Noether fundamental theorem 61
2.7 Hilbert characteristic function 62
2.7.1 Basics on the underlying graded structures 62
2.7.2 First results 67
2.7.3 More advanced steps 68
2.7.4 The formula of van der Waerden 74
2.7.5 Multiplicities galore 77
2.8 Historic note 82
2.8.1 Fractions 82
2.8.2 Prüfer and the determinantal trick 83
2.8.3 Noether and Krull 84
2.8.4 Primary decomposition 85
2.8.5 Hilbert and Artin 86
2.8.6 The Lasker-Noether binary 87
2.8.7 Hilbert function 89
2.9 Exercises 90
3 Overview of module theory 99
3.1 Noetherian modules 99
3.1.1 Chain conditions 99
3.1.2 Composition series 100
3.2 External operations 102
3.3 Free presentation and Fitting ideals 106
3.4 Torsion and torsion-free modules 111
3.5 Historic note 114
3.5.1 Composition series 114
3.5.2 Fitting ideals 115
3.6 Exercises 115
4 Derivations, differentials and Jacobian ideals 119
4.1 Preliminaries 119
4.1.1 Derivations of subalgebras 122
4.1.2 Derivations with values on a larger ring 124
4.2 Differential structures 125
4.2.1 A first structure theorem 125
4.2.2 The universal module of differentials 126
4.2.3 The conormal exact sequence 127
4.2.4 Kähler differentials 130
4.3 The issue of regularity in algebra and geometry 131
4.3.1 The Jacobian ideal 131
4.3.2 Hypersurfaces 132
4.4 Differents and ramification 134
4.4.1 Ramification 134
4.4.2 Purity 136
4.5 Historic note 137
4.6 Exercises 138
Part II
5 Basic advanced theory 145
5.1 Dimension theory 145
5.1.1 Annihilators, 1 145
5.1.2 The Nakayama lemma 145
5.1.3 The Krull dimension and systems of parameters 147
5.2 Associated primes and primary decomposition 151
5.2.1 Annihilators, 2 151
5.2.2 Associated primes 152
5.2.3 Primary decomposition 155
5.3 Depth and Cohen-Macaulay modules 159
5.3.1 Basic properties of depth 162
5.3.2 Mobility of depth 164
5.4 Cohen-Macaulay modules 167
5.4.1 Special properties of Cohen-Macaulay modules 169
5.4.2 Numerical invariants: Gorenstein rings 170
5.5 Historic note 174
5.5.1 Dimension 174
5.5.2 Primary decomposition 174
5.5.3 The depth behind the curtains 175
5.5.4 The KruCheSam theorem 175
5.6 Exercises 176
6 Homological methods 179
6.1 Regular local rings 179
6.1.1 Relation to basic invariants 179
6.1.2 Properties 181
6.2 The homological tool for Noetherian rings 182
6.2.1 Projective modules 182
6.2.2 Homological dimension 184
6.2.3 Chain complexes 200
6.2.4 Basics on derived functors 206
6.2.5 Rees theorem and perfect ideals 222
6.3 The method of the Koszul complex 226
6.3.1 Long exact sequences of Koszul homology 229
6.3.2 The theorem of Serre 234
6.4 Variations on the Koszul complex: determinantal ideals 236
6.4.1 The Eagon-Northcott complex 236
6.4.2 The Scandinavian complex 242
6.4.3 The Japanese-Polish complex 244
6.4.4 The Osnabruck-Recife complex 246
6.5 Historic note 248
6.5.1 Projective modules 248
6.5.2 Homology 249
6.5.3 Injective modules 249
6.5.4 Determinantal ideals 250
6.6 Exercises 251
7 Graded structures 255
7.1 Graded preliminaries 255
7.2 The symmetric algebra 257
7.2.1 Torsion-freeness 258
7.2.2 Ideals of linear type, I 261
7.2.3 Dimension 263
7.3 Rees algebras 269
7.3.1 Geometric roots 269
7.3.2 Dimensions 271
7.3.3 The fiber cone and the analytic spread 276
7.3.4 Ideals of linear type, II 279
7.3.5 Special properties (survey) 284
7.3.6 Specialization 289
7.4 Hilbert function of modules 293
7.4.1 Combinatorial preliminaries 294
7.4.2 The graded Hilbert function 297
7.4.3 Intertwining graded Hilbert functions 303
7.4.4 The local Hilbert-Samuel function 309
7.5 Historic note 316
7.5.1 The Rees algebra 316
7.5.2 The symmetric algebra 316
7.5.3 Artin-Rees lemma 317
7.5.4 Associativity formulas 317
7.6 Exercises 317
Bibliography 321
Index 329