Author | Eisenbud, David. author |
---|---|
Title | Commutative Algebra [electronic resource] : with a View Toward Algebraic Geometry / by David Eisenbud |
Imprint | New York, NY : Springer New York, 1995 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-5350-1 |
Descript | XVI, 788 p. online resource |
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 Henselโs 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 -- 15Grรถbner 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 of Complexes -- 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 and Dimensio -- 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