Commutative Algebra with a View Toward Algebraic Geometry
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,...
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| Format: | eBook |
| Language: | English |
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New York, NY
Springer New York
1995, 1995
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| Edition: | 1st ed. 1995 |
| Series: | Graduate Texts in Mathematics
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- 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
- 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
- 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