Introduction Part I The Algebra-Geometry Lexicon 1 Hilbert's Nullstellensatz 1.1 Maximal Ideals 1.2 Jacobson Rings 1.3 Coordinate Rings Exercises 2 Noetherian and Artinian Rings 2.1 The Noether and Artin Properties for Rings and Modules 2.2 Noetherian Rings and Modules Exercises 3 The Zariski Topology 3.1 Affine Varieties 3.2 Spectra 3.3 Noetherian and Irreducible Spaces Exercises 4 A Summary of the Lexicon 4.1 True Geometry: Affine Varieties 4.2 Abstract Geometry: Spectra Exercises Part II Dimension 5 Krull Dimension and Transcendence Degree Exercises 6 Localization Exercises 7 The Principal Ideal Theorem 7.1 Nal~yama's Lemma and the Principal Ideal Theorem 7.2 The Dimension of Fibers Exercises 8 Integral Extensions 8.1 Integral Closure 8.2 Lying Over, Going Up, and Going Down 8.3 Noether Normalization Exercises Part III Computational Methods 9 Grobner Bases 9.1 Buchberger's Algorithm 9.2 First Application: Elimination Ideals Exercises 10 Fibers and Images of Morphisms Revisited 10.1 The Generic Freeness Lemma 10.2 Fiber Dimension and Constructible Sets 10.3 Application: Invariant Theory Exercises 11 Hilbert Series and Dimension 11.1 The Hilbert-Serre Theorem 11.2 Hilbert Polynomials and Dimension Exercises Part IV Local Rings 12 Dimension Theory
12.1 The Length of a Module 12.2 The Associated Graded Ring Exercises 13 Regular Local Rings 13.1 Basic Properties of Regular Local Rings 13.2 The Jacobian Criterion Exercises 14 Rings of Dimension One 14.1 Regular Rings and Normal Rings 14.2 Multiplicative Ideal Theory 14.3 Dedekind Domains Exercises Solutions of Some Exercises References Notation Index
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