Commutative Algebra: With a View Toward Algebraic Geometry

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Springer Science & Business Media, Mar 30, 1995 - Mathematics - 785 pages
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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.
 

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Contents

Introduction
1
Elementary Definitions
11
Roots of Commutative Algebra
21
Localization
57
Associated Primes and Primary Decomposition
87
Integral Dependence and the Nullstellensatz
117
Filtrations and the ArtinRees Lemma
147
Flat Families
157
Modules of Differentials
385
Homological Methods
421
Depth Codimension and CohenMacaulay Rings
451
Homological Theory of Regular Local Rings
473
Free Resolutions and Fitting Invariants
493
Duality Canonical Modules and Gorenstein Rings
523
Field Theory
561
Multilinear Algebra
571

Completions and Hensels Lemma
181
Dimension Theory
213
Fundamental Definitions of Dimension Theory
227
The Principal Ideal Theorem and Systems of Parameters
233
Dimension and Codimension One
251
Dimension and HilbertSamuel Polynomials
275
The Dimension of Affine Rings
285
Elimination Theory Generic Freeness and the Dimension
307
Grobner Bases
321
Homological Algebra
617
A Sketch of Local Cohomology
691
Category Theory
697
Limits and delimits
705
Where Next?
719
Hints and Solutions for Selected1 Exercises 711
757
Index of Notation
775
Copyright

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