Commutative Algebra With a View Toward Algebraic Geometry,9780387942698

Commutative Algebra With a View Toward Algebraic Geometry

by
ISBN13: 9780387942698
ISBN10: 0387942696
Format: Paperback
Pub. Date: 1994-12-01
Publisher(s): Springer Nature
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Summary

Commutative Algebra is best understood with knowledge of the geometric ideas that have played a great role in its formation, in short, with a view towards algebraic geometry.The author presents a comprehensive view of commutative algebra, from basics, such as localization and primary decomposition, through dimension theory, differentials, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. Many exercises illustrate and sharpen the theory and extended exercises give the reader an active part in complementing the material presented in the text.One novel feature is a chapter devoted to a quick but thorough treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Applications of the theory and even suggestions for computer algebra projects are included.This book will appeal to readers from beginners to advanced students of commutative algebra or algebraic geometry. To help beginners, the essential ideals from algebraic geometry are treated from scratch. Appendices on homological algebra, multilinear algebra and several other useful topics help to make the book relatively self- contained. Novel results and presentations are scattered throughout the text.

Table of Contents

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

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