Commutative Algebra : with a View Toward Algebraic Geometry / by David Eisenbud
(Graduate Texts in Mathematics ; 150)
| データ種別 | 電子ブック |
|---|---|
| 出版者 | New York, NY : Springer New York |
| 出版年 | 1995 |
| 本文言語 | 英語 |
| 大きさ | XVI, 788 p : online resource |
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| 内容注記 | 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 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 |
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| 一般注記 | 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 |
| 著者標目 | *Eisenbud, David author SpringerLink (Online service) |
| 件 名 | LCSH:Mathematics LCSH:Algebraic geometry FREE:Mathematics FREE:Algebraic Geometry |
| 分 類 | DC23:516.35 |
| 巻冊次 | ISBN:9781461253501 
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| ISBN | 9781461253501 |
| URL | http://dx.doi.org/10.1007/978-1-4612-5350-1 |
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