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.

D. EisenbudCommutative Algebra with a View Toward Algebraic Geometry"This text has personality—Those familiar with Eisenbud"s own research will recognize its traces in his choice of topics and manner of approach. The book conveys infectious enthusiasm and the conviction that research in the field is active and yet accessible."—MATHEMATICAL REVIEWS