Commutative algebra with a view toward algebraic geometry

Description

Contents I Basic Constructions II Dimension Theory III Homological Methods Introduction 0 Elementary Definitions 0.1 Rings and Ideals 0.2 Unique Factorization 0.3 Modules Part I Basic Constructions 1 Roots of Commutative Algebra 1.1 Number Theory 1.2 Algebraic Curves and Function Theory 1.3 Invariant Theory 1.4 The Basis Theorem 1.4.1 Finite Generation of Invariants 1.5 Graded Rings 1.6 Algebra and Geometry: The Nullstellensatz 1.7 Geometric Invariant Theory 1.8 Projective Varieties 1.9 Hilbert Functions and Polynomials 1.10 Free Resolutions and the Syzygy Theorem 1.11 Exercises Noetherian Rings and Modules An Analysis of Hilbert's Finiteness Argument Some Rings of Invariants Algebra and Geometry Graded Rings and Projective Geometry Hilbert Functions Free Resolutions Spec, max-Spec, and the Zariski Topology 2 Localization 2.1 Fractions 2.2 Hom and Tensor 2.3 The Construction of Primes 2.4 Rings and Modules of Finite Length 2.5 Products of Domains 2.6 Exercises Z-graded Rings and Their Localizations Partitions of Unity Gluing Constructing Primes Idempotents, Products, and Connected Components 3 Associated Primes and Primary Decomposition 3.1 Associated Primes 3.2 Prime Avoidance 3.3 Primary Decomposition 3.4 Primary Decomposition and Factoriality 3.5 Primary Decomposition in the Graded Case 3.6 Extracting Information from Primary Decomposition 3.7 Why Primary Decomposition Is Not Unique 3.8 Geometric Interpretation of Primary Decomposition 3.9 Symbolic Powers and Functions Vanishing to High Order 3.9.1 A Determinantal Example 3.10 Exercises General Graded Primary Decomposition Primary Decomposition of Monomial Ideals The Question of Uniqueness Determinantal Ideals Total Quotients Prime Avoidance 4 Integral Dependence and the Nullstellensatz 4.1 The Cayley-Hamilton Theorem and Nakayama's Lemma 4.2 Normal Domains and the Normalization Process 4.3 Normalization in the Analytic Case 4.4 Primes in an Integral Extension 4.5 The Nullstellensatz 4.6 Exercises Projective Modules and Locally Free Modules Integral Closure of Ideals Normalization Normalization and Convexity Nullstellensatz Three more Proofs of the Nullstellensatz 5 Filtrations and the Artin-Rees Lemma 5.1 Associated Graded Rings and Modules 5.2 The Blowup Algebra 5.3 The Krull Intersection Theorem 5.4 The Tangent Case 5.5 Exercises 6 Flat Families 6.1 Elementary Examples 6.2 Introduction to Tor 6.3 Criteria for Flatness 6.4 The Local Criterion for Flatness 6.5 The Rees Algebra 6.6 Exercises Flat Families of Graded Modules Embedded First-Order Deformations 7 Completions and Hensel's Lemma 7.1 Examples and Definitions 7.2 The Utility of Completions 7.3 Lifting Idempotents 7.4 Cohen Structure Theory and Coefficient Fields 7.5 Basic Properties of Completion 7.6 Maps from Power Series Rings 7.7 Exercises Modules Whose Completions Are Isomorphic The Krull Topology and Cauchy Sequences Completions from Power Series Coefficient Fields Other Versions of Hensel's Lemma Part II Dimension Theory 8 Introduction to Dimension Theory 8.1 Axioms for Dimension 8.2 Other Characterizations of Dimension 8.2.1 Affine Rings and Noether Normalization 8.2.2 Systems of Parameters and Krull's Principal Ideal Theorem 8.3.3 The Degree of the Hilbert Polynomial 9 Fundamental Definitions of Dimension Theory 9.1 Dimension Zero 9.2 Exercises 10 The Principal Ideal Theorem and Systems of Parameters 10.1 Systems of Parameters and Parameter Ideals 10.2 Dimension of Base and Fiber 10.3 Regular Local Rings 10.4 Exercises Determinantal Ideals Hilbert Series of a Graded Module 11 Dimension and Codimension One 11.1 Discrete Valuation Rings 11.2 Normal Rings and Serre's Criterion 11.3 Invertible Modules 11.4 Unique Factorization of Codimension-One Ideals 11.5 Divisors and Multiplicities 11.6 Multiplicity of Principal Ideals 11.7 Exercises Valuation Rings The Grothendieck Ring 12 Dimension and Hilbert-Samuel Polynomials 12.1 Hilbert-Samuel Functions 12.2 Exercises Analytic Spread and the Fiber of a Blowup Multiplicities Hilbert Series 13 The Dimension of Affine Rings 13.1 Noether Normalization 13.2 The Nullstellensatz 13.3 Finiteness of the Integral Closure 13.4 Exercises Quotients by Finite Groups Primes in Polynomials Rings Dimension in the Graded Case Noether Normalization in the Complete Case Products and Reduction to the Diagonal Equational Characterization of Systems of Parameters 14 Elimination Theory, Generic Freeness, and the Dimension of Fibers 14.1 Elimination Theory 14.2 Generic Freeness 14.3 The Dimension of Fibers 14.4 Exercises Elimination Theory 15 Gröbner Bases 15.1 Monomials and Terms 15.1.1 Hilbert Function and Polynomial 15.1.2 Syzygies of Monomial Submodules 15.2 Monomial Orders 15.3 The Division Algorithm 15.4 Gröbner Bases 15.5 Syzygies 15.6 History of Gröbner Bases 15.7 A Property of Reverse Lexicographic Order 15.8 Gröbner Bases and Flat Families 15.9 Generic Initial Ideals 15.9.1 Existence of the Generic Initial Ideal 15.9.2 The Generic Initial Ideal is Borel-Fixed 15.10 Applications 15.10.1 Ideal Membership 15.10.2 Hilbert Function and Polynomial 15.10.3 Associated Graded Ring 15.10.4 Elimination 15.10.5 Projective Closure and Ideal at Infinity 15.10.6 Saturation 15.10.7 Lifting Homomorphisms 15.10.8 Syzygies and Constructive Module Theory 15.10.9 What's Left? 15.11 Exercises 15.12 Appendix: Some Computer Algebra Projects Project 1. Zero-Dimensional Gorenstein Ideals Project 2. Factoring Out a General Element from an sth Syzygy Project 3. Resolutions over Hypersurfaces Project 4. Rational Curves of Degree r + 1 in R^r Project 5. Regularity of Rational Curves Project 6. Some Monomial Curve Singularities Project 7. Some Interesting Prime Ideals 16 Modules of Differentials 16.1 Computation of Differentials 16.2 Differentials and the Cotangent Bundle 16.3 Colimits and Localization 16.4 Tangent Vector Fields and Infinitesimal Morphisms 16.5 Differentials and Field Extensions 16.6 Jacobian Criterion for Regularity 16.7 Smoothness and Generic Smoothness 16.8 Appendix: Another Construction of Kähler Differentials 16.9 Exercises Part III Homological Methods 17 Regular Sequences and the Koszul Complex 17.1 Koszul Complexes of Lengths 1 and 2 17.2 Koszul Complexes in General 17.3 Building the Koszul Complex from Parts 17.4 Duality and Homotopies 17.5 The Koszul Complex and the Cotangent Bundle of Projective Space 17.6 Exercises Free Resolutions of Monomial Ideals Conormal Sequence of a Complete Intersection Regular Sequences Are Like Sequences of Variables Blowup Algebra and Normal Cone of a Regular Sequence Geometric Contexts of the Koszul Complex 18 Depth, Codimension, and Cohen-Macaulay Rings 18.1 Depth 18.1.1 Depth and the Vanishing of Ext 18.2 Cohen-Macaulay Rings 18.3 Proving Primeness with Serre's Criterion 18.4 Flatness and Depth 18.5 Some Examples 18.6 Exercises 19 Homological Theory of Regular Local Rings 19.1 Projective Dimension and Minimal Resolutions 19.2 Global Dimension and the Syzygy Theorem 19.3 Depth and Projective Dimension: The Auslander-Buchsbaum Formula 19.4 Stably Free Modules and Factoriality of Regular Local Rings 19.5 Exercises Regular Rings Modules over a Dedekind Domain The Auslander-Buchsbaum Formula Projective Dimension and Cohen-Macaulay Rings Hilbert Function and Grothendieck Group The Chern Polynomial 20 Free Resolutions and Fitting Invariants 20.1 The Uniqueness of Free Resolutions 20.2 Fitting Ideals 20.3 What Makes a Complex Exact? 20.4 The Hilbert-Burch Theorem 20.4.1 Cubic Surfaces and Sextuples of Points in the Plane 20.5 Castelnuovo-Mumford Regularity 20.5.1 Regularity and Hyperplane Sections 20.5.2 Regularity of Generic Initial Ideals 20.5.3 Historical Notes on Regularity 20.6 Exercises Fitting Ideals and the Structure of Modules Projectives of Constant Rank Castelnuovo-Mumford Regularity 21 Duality, Canonical Modules, and Gorenstein Rings 21.1 Duality for Modules of Finite Length 21.2 Zero-Dimensional Gorenstein Rings 21.3 Canonical Modules and Gorenstein Rings In Higher Dimension 21.4 Maximal Cohen-Macaulay Modules 21.5 Modules of Finite Injective Dimension 21.6 Uniqueness and (Often) Existence 21.7 Localization and Completion of the Canonical Module 21.8 Complete Intersections and Other Gorenstein Rings 21.9 Duality for Maximal Cohen-Macaulay Modules 21.10 Linkage 21.11 Duality in the Graded Case 21.12 Exercises The Zero-Dimensional Case and Duality Higher Dimension The Canonical Module as Ideal Linkage and the Cayley-Bacharach Theorem Appendix 2 Multilinear Algebra A2.2 Tensor Products Hints and Solutions for Selected Exercises Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Reference Index of Notation Index