Commutative Algebra

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. Language: English