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Tables of Contents for Commutative Algebra With a View Toward Algebraic Geometry
Chapter/Section Title
Page #
Page Count
Introduction
1
18
Advice for the Beginner
2
1
Information for the Expert
2
4
Prerequisites
6
1
Sources
6
1
Courses
7
2
A First Course
7
1
A Second Course
8
1
Acknowledgements
9
2
Elementary Definitions
11
8
Rings and Ideals
11
2
Unique Factorization
13
2
Modules
15
4
I Basic Constructions
19
194
Roots of Commutative Algebra
21
36
Number Theory
21
2
Algebraic Curves and Function Theory
23
1
Invariant Theory
24
3
The Basis Theorem
27
3
Finite Generation of Invariants
29
1
Graded Rings
30
1
Algebra and Geometry: The Nullstellensatz
31
6
Geometric Invariant Theory
37
2
Projective Varieties
39
3
Hilbert Functions and Polynomials
42
2
Free Resolutions and the Syzygy Theorem
44
2
Exercises
46
11
Noetherian Rings and Modules
46
1
An Analysis of Hilbert's Finiteness Argument
47
1
Some Rings of Invariants
48
1
Algebra and Geometry
49
3
Graded Rings and Projective Geometry
52
1
Hilbert Functions
53
1
Free Resolutions
54
1
Spec, max-Spec, and the Zariski Topology
54
3
Localization
57
30
Fractions
59
3
Home and Tensor
62
8
The Construction of Primes
70
1
Rings and Modules of Finite Length
71
7
Products of Domains
78
1
Exercises
78
9
Z-graded Rings and Their Localizations
81
2
Partitions of Unity
83
1
Gluing
83
1
Constructing Primes
84
1
Idempotents, Products, and Connected Components
85
2
Associated Primes and Primary Decomposition
87
30
Associated Primes
89
1
Prime Avoidance
90
4
Primary Decomposition
94
4
Primary Decomposition and Factoriality
98
1
Primary Decomposition in the Graded Case
99
1
Extracting Information from Primary Decomposition
100
2
Why Primary Decomposition Is Not Unique
102
1
Geometric Interpretation of Primary Decomposition
103
2
Symbolic Powers and Functions Vanishing to High Order
105
4
A Determinantal Example
107
2
Exercises
109
8
General Graded Primary Decomposition
110
1
Primary Decomposition of Monomial Ideals
111
1
The Question of Uniqueness
112
1
Determinantal Ideals
113
1
Total Quotients
113
1
Prime Avoidance
114
3
Integral Dependence and the Nullstellensatz
117
30
The Cayley-Hamilton Theorem and Nakayama's Lemma
119
6
Normal Domains and the Normalization Process
125
3
Normalization in the Analytic Case
128
1
Primes in an Integral Extension
129
2
The Nullstellensatz
131
4
Exercises
135
12
Nakayama's Lemma
136
1
Projective Modules and Locally Free Modules
136
1
Integral Closure of Ideals
137
1
Normalization
138
1
Normalization and Convexity
139
3
Nullstellensatz
142
1
Three More Proofs of the Nullstellensatz
142
5
Filtrations and the Artin-Rees Lemma
147
10
Associated Graded Rings and Modules
148
2
The Blowup Algebra
150
2
The Krull Intersection Theorem
152
1
The Tangent Cone
153
1
Exercises
154
3
Flat Families
157
24
Elementary Examples
159
2
Introduction to Tor
161
1
Criteria for Flatness
162
5
The Local Criterion for Flatness
167
4
The Rees Algebra
171
1
Exercises
172
9
Flat Families of Graded Modules
175
1
Embedded First-Order Deformations
176
5
Completions and Hensel's Lemma
181
32
Examples and Definitions
181
3
The Utility of Completions
184
4
Lifting Idempotents
188
3
Cohen Structure Theory and Coefficient Fields
191
3
Basic Properties of Completion
194
6
Maps from Power Series Rings
200
5
Exercises
205
8
Modules Whose Completions Are Isomorphic
205
1
The Krull Topology and Cauchy Sequences
206
1
Completions from Power Series
207
1
Coefficient Fields
207
1
Other Versions of Hensel's Lemma
208
5
II Dimension Theory
213
208
Introduction to Dimension Theory
215
12
Axioms for Dimension
220
2
Other Characterizations of Dimension
222
5
Affine Rings and Noether Normalization
223
1
Systems of Parameters and Krull's Principal Ideal Theorem
224
1
The Degree of the Hilbert Polynomial
225
2
Fundamental Definitions of Dimension Theory
227
6
Dimension Zero
229
1
Exercises
230
3
The Principal Ideal Theorem and Systems of Parameters
233
18
Systems of Parameters and Ideals of Finite Colength
236
2
Dimension of Base and Fiber
238
4
Regular Local Rings
242
2
Exercises
244
7
Determinantal Ideals
246
1
Hilbert Series of a Graded Module
247
4
Dimension and Codimension One
251
24
Discrete Valuation Rings
251
2
Normal Rings and Serre's Criterion
253
4
Invertible Modules
257
3
Unique Factorization of Codimension-One Ideals
260
2
Divisors and Multiplicities
262
3
Multiplicity of Principal Ideals
265
3
Exercises
268
7
Valuation Rings
268
1
The Grothendieck Ring
269
6
Dimension and Hilbert-Samuel Polynomials
275
10
Hilbert-Samuel Functions
276
3
Exercises
279
6
Analytic Spread and the Fiber of a Blowup
280
1
Multiplicities
280
4
Hilbert Series
284
1
The Dimension of Affine Rings
285
22
Noether Normalization
285
11
The Nullstellensatz
296
1
Finiteness of the Integral Closure
297
3
Exercises
300
7
Quotients by Finite Groups
300
1
Primes in Polynomial Rings
301
1
Dimension in the Graded Case
302
1
Noether Normalization in the Complete Case
303
1
Products and Reduction to the Diagonal
304
2
Equational Characterization of Systems of Parameters
306
1
Elimination Theory, Generic Freeness, and the Dimension of Fibers
307
14
Elimination Theory
307
5
Generic Freeness
312
1
The Dimension of Fibers
313
5
Exercises
318
3
Elimination Theory
318
3
Grobner Bases
321
64
Constructive Module Theory
322
1
Elimination Theory
322
1
Monomials and Terms
323
4
Hilbert Function and Polynomial
324
2
Syzygies of Monomial Submodules
326
1
Monomial Orders
327
6
The Division Algorithm
333
2
Grobner Bases
335
2
Syzygies
337
3
History of Grobner Bases
340
2
A Property of Reverse Lexicographic Order
342
3
Grobner Bases and Flat Families
345
6
Generic Initial Ideals
351
7
Existence of the Generic Initial Ideal
353
1
The Generic Initial Ideal is Borel-Fixed
354
1
The Nature of Borel-Fixed Ideals
355
3
Applications
358
10
Ideal Membership
359
1
Hilbert Function and Polynomial
359
1
Associated Graded Ring
360
1
Elimination
361
1
Projective Closure and Ideal at Infinity
362
1
Saturation
363
1
Lifting Homomorphisms
364
1
Syzygies and Constructive Module Theory
365
2
What's Left?
367
1
Exercises
368
10
Appendix: Some Computer Algebra Projects
378
7
Project 1. Zero-dimensional Gorenstein Ideals
376
1
Project 2. Factoring Out a General Element from an sth Syzygy
377
1
Project 3. Resolutions over Hypersurfaces
377
1
Project 4. Rational Curves of Degree r + 1 in Pr
378
1
Project 5. Regularity of Rational Curves
378
1
Project 6. Some Monomial Curve Singularities
379
1
Project 7. Some Interesting Prime Ideals
379
6
Modules of Differentials
385
36
Computation of Differentials
390
1
Differentials and the Cotangent Bundle
390
3
Colimits and Localization
393
5
Tangent Vector Fields and Infinitesimal Morphisms
398
2
Differentials and Field Extensions
400
4
Jacobian Criterion for Regularity
404
3
Smoothness and Generic Smoothness
407
3
Appendix: Another Construction of Kahler Differentials
410
2
Exercises
412
9
III Homological Methods
421
134
Regular Sequences and the Koszul Complex
423
28
Koszul Complexes of Lengths 1 and 2
424
3
Koszul Complexes in General
427
4
Building the Koszul Complex from Parts
431
5
Duality and Homotopies
436
4
The Koszul Complex and the Cotangent Bundle of Projective Space
440
1
Exercises
441
10
Free Resolutions of Monomial Ideals
443
1
Conormal Sequence of a Complete Intersection
444
1
Regular Sequences Are Like Sequences of Variables
445
1
Blowup Algebra and Normal Cone of a Regular Sequence
445
2
Geometric Contexts of the Koszul Complex
447
4
Depth, Codimension, and Cohen-Macaulay Rings
451
22
Depth
451
4
Depth and the Vanishing of Ext
453
2
Cohen-Macaulay Rings
455
6
Proving Primeness with Serre's Criterion
461
3
Flatness and Depth
464
2
Some Examples
466
3
Exercises
469
4
Homological Theory of Regular Local Rings
473
20
Projective Dimension and Minimal Resolutions
473
5
Global Dimension and the Syzygy Theorem
478
1
Depth and Projective Dimension: The Auslander-Buchsbaum Formula
479
5
Stably Free Modules and Factoriality of Regular Local Rings
484
4
Exercises
488
5
Regular Rings
488
1
Modules over a Dedekind Domain
488
1
The Auslander-Buchsbaum Formula
489
1
Projective Dimension and Cohen-Macaulay Rings
489
1
Hilbert Function and Grothendieck Group
490
2
The Chern Polynomial
492
1
Free Resolutions and Fitting Invariants
493
30
The Uniqueness of Free Resolutions
494
2
Fitting Ideals
496
4
What Makes a Complex Exact?
500
6
The Hilbert-Burch Theorem
506
3
Cubic Surfaces and Sextuples of Points in the Plane
508
1
Castelnuovo-Mumford Regularity
509
6
Regularity and Hyperplane Sections
513
1
Regularity of Generic Initial Ideals
514
1
Historical Notes on Regularity
514
1
Exercises
515
8
Fitting Ideals and the Structure of Modules
515
3
Projectives of Constant Rank
518
3
Castelnuovo-Mumford Regularity
521
2
Duality, Canonical Modules, and Gorenstein Rings
523
32
Duality for Modules of Finite Length
524
5
Zero-Dimensional Gorenstein Rings
529
3
Canonical Modules and Gorenstein Rings in Higher Dimension
532
1
Maximal Cohen-Macaulay Modules
533
1
Modules of Finite Injective Dimension
534
4
Uniqueness and (Often) Existence
538
2
Localization and Completion of the Canonical Module
540
1
Complete Intersections and Other Gorenstein Rings
541
1
Duality for Maximal Cohen-Macaulay Modules
542
1
Linkage
543
6
Duality in the Graded Case
549
1
Exercises
550
5
The Zero-Dimensional Case and Duality
550
2
Higher Dimension
552
3
The Canonical Module as Ideal
555
1
Linkage and the Cayley-Bacharach Theorem
556
Appendix 1 Field Theory
555
10
A1.1 Transcendence Degree
561
2
A1.2 Separability
563
2
A1.3 p-Bases
565
1
A1.3.1 Exercises
568
Appendix 2 Multilinear Algebra
565
46
A2.1 Introduction
571
2
A2.2 Tensor Products
573
1
A2.3 Symmetric and Exterior Algebras
574
7
A2.3.1 Bases
578
2
A2.3.2 Exercises
580
1
A2.4 Coalgebra Structures and Divided Powers
581
9
A2.4.1 S(M)* and S(M) as Modules over One Another
582
8
A2.5 Schur Functors
590
6
A2.5.1 Exercises
594
2
A2.6 Complexes Constructed by Multilinear Algebra
596
15
A2.6.1 Strands of the Koszul Complex
597
12
A2.6.2 Exercises
609
2
Appendix 3 Homological Algebra
611
3
A3.1 Introduction
617
Part I: Resolutions and Derived Functors
614
36
A3.2 Free and Projective Modules
621
2
A3.3 Free and Projective Resolutions
623
1
A3.4 Injective Modules and Resolutions
624
8
A3.4.1 Exercises
630
1
Injective Envelopes
630
1
Injective Modules over Noetherian Rings
630
2
A3.5 Basic Constructions with Complexes
632
1
A3.5.1 Notation and Definitions
632
1
A3.6 Maps and Homotopies of Complexes
633
4
A3.7 Exact Sequences of Complexes
637
2
A3.7.1 Exercises
638
1
A3.8 The Long Exact Sequence in Homology
639
4
A3.8.1 Exercises
640
1
Diagrams and Syzygies
640
3
A3.9 Derived Functors
643
3
A3.9.1 Exercise on Derived Functors
645
1
A3.10 Tor
646
3
A3.10.1 Exercises: Tor
646
3
A3.11 Ext
649
1
A3.11.1 Exercises: Ext
651
5
A3.11.2 Local Cohomology
656
Part II: From Mapping Cones to Spectral Sequences
650
33
A3.12 The Mapping Cone and Double Complexes
656
7
A3.12.1 Exercises: Mapping Cones and Double Complexes
660
3
A3.13 Spectral Sequences
663
21
A3.13.1 Mapping Cones Revisited
664
1
A3.13.2 Exact Couples
665
3
A3.13.3 Filtered Differential Modules and Complexes
668
3
A3.13.4 The Spectral Sequence of a Double Complex
671
6
A3.13.5 Exact Sequence of Terms of Low Degree
677
1
A3.13.6 Exercises on Spectral Sequences
678
6
A3.14 Derived Categories
684
A3.14.1 Step One: The Homotopy Category of Complexes
685
1
A3.14.2 Step Two: The Derived Category
686
2
A3.14.3 Exercises on the Derived Category
688
Appendix 4 A Sketch of Local Cohomology
683
6
A4.1 Local Cohomology and Global Cohomology
693
1
A4.2 Local Duality
694
1
A4.3 Depth and Dimension
695
Appendix 5 Category Theory
689
8
A5.1 Categories, Functors, and Natural Transformations
697
2
A5.2 Adjoint Functors
699
4
A5.2.1 Uniqueness
700
1
A5.2.2 Some Examples
700
1
A5.2.3 Another Characterization of Adjoints
701
1
A5.2.4 Adjoints and Limits
702
1
A5.3 Representable Functors and Yoneda's Lemma
703
Appendix 6 Limits and Colimits
697
12
A6.1 Colimits in the Category of Modules
708
3
A6.2 Flat Modules as Colimits of Free Modules
711
2
A6.3 Colimits in the Category of Commutative Algebras
713
2
A6.4 Exercises
715
Appendix 7 Where Next?
709
2
Hints and Solutions for Selected1 Exercises
711
46
References
757
18
Index of Notation
775
4
Index
779