Table of Contents
Introduction 1
Part I The Algebra-Geometry Lexicon
1 Hilbert's Nullstellensatz 7
1.1 Maximal Ideals 8
1.2 Jacobson Rings 12
1.3 Coordinate Rings 16
Exercises 19
2 Noetherian and Artinian Rings 23
2.1 The Noether and Artin Properties for Rings and Modules 23
2.2 Noetherian Rings and Modules 28
Exercises 30
3 The Zariski Topology 33
3.1 Affine Varieties 33
3.2 Spectra 36
3.3 Noetherian and Irreducible Spaces 38
Exercises 42
4 A Summary of the Lexicon 45
4.1 True Geometry: Affine Varieties 45
4.2 Abstract Geometry: Spectra 46
Exercises 48
Part II Dimension
5 Krull Dimension and Transcendence Degree 51
Exercises 60
6 Localization 63
Exercises 70
7 The Principal Ideal Theorem 75
7.1 Nakayama's Lemma and the Principal Ideal Theorem 75
7.2 The Dimension of Fibers 81
Exercises 87
8 Integral Extensions 93
8.1 Integral Closure 93
8.2 Lying Over, Going Up, and Going Down 99
8.3 Noether Normalization 104
Exercises 111
Part III Computational Methods
9 Gröbner Bases 117
9.1 Buchberger's Algorithm 118
9.2 First Application: Elimination Ideals 127
Exercises 133
10 Fibers and Images of Morphisms Revisited 137
10.1 The Generic Freeness Lemma 137
10.2 Fiber Dimension and Constructible Sets 142
10.3 Application: Invariant Theory 144
Exercises 148
11 Hilbert Series and Dimension 151
11.1 The Hilbert-Serre Theorem 151
11.2 Hilbert Polynomials and Dimension 157
Exercises 161
Part IV Local Rings
12 Dimension Theory 167
12.1 The Length of a Module 167
12.2 The Associated Graded Ring 170
Exercises 176
13 Regular Local Rings 181
13.1 Basic Properties of Regular Local Rings 181
13.2 The Jacobian Criterion 185
Exercises 193
14 Rings of Dimension One 197
14.1 Regular Rings and Normal Rings 197
14.2 Multiplicative Ideal Theory 201
14.3 Dedekind Domains 206
Exercises 212
Solutions of Some Exercises 217
References 235
Notation 239
Index 241
