Commutative algebra with a view toward algebraic geometry

✍ Scribed by David Eisenbud

• Publisher: Springer-Verlag New York, Inc.
• Year: 1995
• Tongue: English
• Leaves: 784
• Series: Graduate Texts in Mathematics; v. 150
• Edition: 1
• Category: Library

✦ Table of Contents

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