Commutative Algebra

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.

Advice for the Beginner. - Information for the Expert. - Prerequisites. - Sources. - Courses. - Acknowledgements. - 0 Elementary Definitions. - 0. 1 Rings and Ideals. - 0. 2 Unique Factorization. - 0. 3 Modules. - I Basic Constructions. - 1 Roots of Commutative Algebra. - 2 Localization. - 3 Associated Primes and Primary Decomposition. - 4 Integral Dependence and the Nullstellensatz. - 5 Filtrations and the Artin-Rees Lemma. - 6 Flat Families. - 7 Completions and Hensel s Lemma. - II Dimension Theory. - 8 Introduction to Dimension Theory. - 9 Fundamental Definitions of Dimension Theory. - 10 The Principal Ideal Theorem and Systems of Parameters. - 11 Dimension and Codimension One. - 12 Dimension and Hilbert-Samuel Polynomials. - 13 The Dimension of Affine Rings. - 14 Elimination Theory, Generic Freeness, and the Dimension of Fibers. - 15Gröbner Bases. - 16 Modules of Differentials. - III Homological Methods. - 17 Regular Sequences and the Koszul Complex. - 18 Depth, Codimension, and Cohen-Macaulay Rings. - 19 Homological Theory of Regular Local Rings. - 20 Free Resolutions and Fitting Invariants. - 21 Duality, Canonical Modules, and Gorenstein Rings. - Appendix 1 Field Theory. - A1. 1 Transcendence Degree. - A1. 2 Separability. - A1. 3. 1 Exercises. - Appendix 2 Multilinear Algebra. - A2. 1 Introduction. - A2. 2 Tensor Product. - A2. 3 Symmetric and Exterior Algebras. - A2. 3. 1 Bases. - A2. 3. 2 Exercises. - A2. 4 Coalgebra Structures and Divided Powers. - A2. 5 Schur Functors. - A2. 5. 1 Exercises. - A2. 6 Complexes Constructed by Multilinear Algebra. - A2. 6. 1 Strands of the Koszul Comple. - A2. 6. 2 Exercises. - Appendix 3 Homological Algebra. - A3. 1 Introduction. - I: Resolutions and Derived Functors. - A3. 2 Free and Projective Modules. - A3. 3 Free and Projective Resolutions. - A3. 4 Injective Modules and Resolutions. - A3. 4. 1 Exercises. - Injective Envelopes. - Injective Modules over Noetherian Rings. - A3. 5 Basic Constructions with Complexes. - A3. 5. 1 Notation and Definitions. - A3. 6 Maps and Homotopies of Complexes. - A3. 7 Exact Sequences ofComplexes. - A3. 7. 1 Exercises. - A3. 8 The Long Exact Sequence in Homology. - A3. 8. 1 Exercises. - Diagrams and Syzygies. - A3. 9 Derived Functors. - A3. 9. 1 Exercise on Derived Functors. - A3. 10 Tor. - A3. 10. 1 Exercises: Tor. - A3. 1l Ext. - A3. 11. 1 Exercises: Ext. - A3. 11. 2 Local Cohomology. - II: From Mapping Cones to Spectral Sequences. - A3. 12 The Mapping Cone and Double Complexe. - A3. 12. 1 Exercises: Mapping Cones and Double Complexes. - A3. 13 Spectral Sequences. - A3. 13. 1 Mapping Cones Revisited. - A3. 13. 2 Exact Couples. - A3. 13. 3 Filtered Differential Modules and Complexes. - A3. 13. 4 The Spectral Sequence of a Double Complex. - A3. 13. 5 Exact Sequence of Terms of Low Degree. - A3. 13. 6 Exercises on Spectral Sequences. - A3. 14 Derived Categories. - A3. 14. 1 Step One: The Homotopy Category of Complexes. - A3. 14. 2 Step Two: The Derived Category. - A3. 14. 3 Exercises on the Derived Category. - Appendix 4 A Sketch of Local Cohomology. - A4. 1 Local Cohomology and Global Cohomology. - A4. 2 Local Duality. - A4. 3 Depth andDimensio. - Appendix 5 Category Theory. - A5. 1 Categories, Functors, and Natural Transformations. - A5. 2 Adjoint Functors. - A5. 2. 1 Uniqueness. - A5. 2. 2 Some Examples. - A5. 2. 3 Another Characterization of Adjoints. - A5. 2. 4 Adjoints and Limits. - A5. 3 Representable Functors and Yoneda's Lemma. - Appendix 6 Limits and Colimits. - A6. 1 Colimits in the Category of Modules. - A6. 2 Flat Modules as Colimits of Free Modules. - A6. 3 Colimits in the Category of Commutative Algebras. - A6. 4 Exercises. - Appendix 7 Where Next? . - References. - Index of Notation.