This textbook offers a thorough, modern introduction into commutative algebra. It is intented mainly to serve as a guide for a course of one or two semesters, or for self-study. The carefully selected subject matter concentrates on the concepts and results at the center of the field. The book maintains a constant view on the natural geometric context, enabling the reader to gain a deeper understanding of the material. Although it emphasizes theory, three chapters are devoted to computational aspects. Many illustrative examples and exercises enrich the text.
Author(s): Gregor Kemper
Series: Graduate Texts in Mathematics
Edition: draft
Publisher: Springer
Year: 2009
Language: English
Pages: 320
Introduction......Page 9
Part I The Algebra Geometry Lexicon......Page 15
Hilbert's Nullstellensatz......Page 17
Maximal Ideals......Page 18
Jacobson Rings......Page 22
Coordinate Rings......Page 26
Exercises......Page 29
The Noether and Artin Property for Rings and Modules......Page 33
Noetherian Rings and Modules......Page 38
Exercises......Page 40
Affine Varieties......Page 43
Spectra......Page 46
Noetherian and Irreducible Spaces......Page 48
Exercises......Page 52
True Geometry: Affine Varieties......Page 55
Abstract Geometry: Spectra......Page 56
Exercises......Page 58
Part II Dimension......Page 59
Krull Dimension and Transcendence Degree......Page 61
Exercises......Page 70
Localization......Page 73
Exercises......Page 80
Nakayama's Lemma and the Principal Ideal Theorem......Page 85
The Dimension of Fibers......Page 91
Exercises......Page 97
Integral Closure......Page 103
Lying Over, Going Up and Going Down......Page 109
Noether Normalization......Page 114
Exercises......Page 121
Part III Computational Methods......Page 125
Gröbner Bases......Page 127
Buchberger's Algorithm......Page 128
First Application: Elimination Ideals......Page 137
Exercises......Page 143
The Generic Freeness Lemma......Page 147
Fiber Dimension and Constructible Sets......Page 152
Application: Invariant Theory......Page 154
Exercises......Page 158
The Hilbert-Serre Theorem......Page 161
Hilbert Polynomials and Dimension......Page 167
Exercises......Page 171
Part IV Local Rings......Page 175
The Length of a Module......Page 177
The Associated Graded Ring......Page 180
Exercises......Page 186
Basic Properties of Regular Local Rings......Page 191
The Jacobian Criterion......Page 195
Exercises......Page 203
Regular Rings and Normal Rings......Page 207
Multiplicative Ideal Theory......Page 211
Dedekind Domains......Page 216
Exercises......Page 222
Solutions of Exercises......Page 227
References......Page 309
Notation......Page 313
Index......Page 315
